topology, measure theory and banach spaces semevi...b. c and j. rodr´ıguez, birkhoff integral for...
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![Page 1: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/1.jpg)
Contents
Topology, measure theory and Banach spaces
B. Cascales
Universidad de Murcia
Second Meeting on Vector Measures and Integration.Sevilla, November 16/18 2006
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Contents
The co-authors
B. C and G. Godefroy, Angelicity and the boundary problem,Mathematika 45 (1998), no. 1, 105–112. MR 99f:46019
B. C, V. Kadets, and J. Rodrıguez, The pettis integral for multi-valuedfunctions via single-valued ones, J. Math. Anal. Appl. (2006). To appear.
B. C, G. Manjabacas, and G. Vera, A Krein-Smulian type result inBanach spaces, Quart. J. Math. Oxford Ser. (2) 48 (1997), no. 190,161–167. MR 99c:46009
B. C and J. Rodrıguez, Birkhoff integral for multi-valued functions, J.Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special issue dedicated toJohn Horvath. MR MR2088679
B. C and J. Rodrıguez, The Birkhoff integral and the property ofBourgain, Math. Ann. 331 (2005), no. 2, 259–279. MR (2006i:28006)
B. C and R. Shvydkoy, On the Krein-Smulian theorem for weakertopologies, Illinois J. Math. 47 (2003), no. 4, 957–976. MR(2004m:46044)
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Contents
1 Bourgain property and compactness with respect to boundaries
2 Bourgain property and Birkhoff integrability
3 Aumman&Debreu&Pettis integrals multifunctions
![Page 4: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/4.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
Throughout the lecture. . .
X is a Banach space equipped with its norm ‖ ‖;K is a Hausdorff compact and C(K) is equipped with its supremum norm.
A subset B ⊂ BX ∗= x∗ ∈ X ∗ ; ‖ x∗ ‖≤ 1 is a boundary forBX ∗ if for any x ∈ X , there is x∗ ∈ B such that x∗(x) =‖ x ‖ .
A simple example of boundary is provided by Ext (BX ∗) theset of extreme points of BX ∗ .
e1e2
e3
![Page 5: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/5.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
Throughout the lecture. . .
X is a Banach space equipped with its norm ‖ ‖;K is a Hausdorff compact and C(K) is equipped with its supremum norm.
A subset B ⊂ BX ∗= x∗ ∈ X ∗ ; ‖ x∗ ‖≤ 1 is a boundary forBX ∗ if for any x ∈ X , there is x∗ ∈ B such that x∗(x) =‖ x ‖ .
A simple example of boundary is provided by Ext (BX ∗) theset of extreme points of BX ∗ .
e1e2
e3
![Page 6: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/6.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
Throughout the lecture. . .
X is a Banach space equipped with its norm ‖ ‖;K is a Hausdorff compact and C(K) is equipped with its supremum norm.
A subset B ⊂ BX ∗= x∗ ∈ X ∗ ; ‖ x∗ ‖≤ 1 is a boundary forBX ∗ if for any x ∈ X , there is x∗ ∈ B such that x∗(x) =‖ x ‖ .
A simple example of boundary is provided by Ext (BX ∗) theset of extreme points of BX ∗ .
e1e2
e3
![Page 7: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/7.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
Throughout the lecture. . .
X is a Banach space equipped with its norm ‖ ‖;K is a Hausdorff compact and C(K) is equipped with its supremum norm.
A subset B ⊂ BX ∗= x∗ ∈ X ∗ ; ‖ x∗ ‖≤ 1 is a boundary forBX ∗ if for any x ∈ X , there is x∗ ∈ B such that x∗(x) =‖ x ‖ .
A simple example of boundary is provided by Ext (BX ∗) theset of extreme points of BX ∗ .
e1e2
e3
![Page 8: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/8.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
The boundary problem (Godefroy)...extremal test for compactness
Let X Banach space, B ⊂ BX ∗ boundary and denote by τp(B) thetopology defined on X by the pointwise convergence on B. Let Hbe a norm bounded and τp(B)-compact subset of X .
Is H weakly compact?
Pioneer positive known results:
1 1952, Grothendieck: X = C (K ) y B = Ext(BC(K)∗);2 1963, Rainwater: B = Ext(BX ∗), H τp(B)-seq.compact;3 1972, James: BX ⊂ BX ∗∗ boundary, characterization of
reflexivity;4 1972, Simons: H τp(B)-seq.compact and B any boundary;5 1974, de Wilde: H convex and B any boundary;6 1982, Bourgain-Talagrand: B = Ext(BX ∗), arbitrary H.
![Page 9: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/9.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
The boundary problem (Godefroy)...extremal test for compactness
Let X Banach space, B ⊂ BX ∗ boundary and denote by τp(B) thetopology defined on X by the pointwise convergence on B. Let Hbe a norm bounded and τp(B)-compact subset of X .
Is H weakly compact?
Pioneer positive known results:
1 1952, Grothendieck: X = C (K ) y B = Ext(BC(K)∗);2 1963, Rainwater: B = Ext(BX ∗), H τp(B)-seq.compact;3 1972, James: BX ⊂ BX ∗∗ boundary, characterization of
reflexivity;4 1972, Simons: H τp(B)-seq.compact and B any boundary;5 1974, de Wilde: H convex and B any boundary;6 1982, Bourgain-Talagrand: B = Ext(BX ∗), arbitrary H.
![Page 10: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/10.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
The boundary problem (Godefroy)...extremal test for compactness
Let X Banach space, B ⊂ BX ∗ boundary and denote by τp(B) thetopology defined on X by the pointwise convergence on B. Let Hbe a norm bounded and τp(B)-compact subset of X .
Is H weakly compact?
Pioneer positive known results:
1 1952, Grothendieck: X = C (K ) y B = Ext(BC(K)∗);
2 1963, Rainwater: B = Ext(BX ∗), H τp(B)-seq.compact;3 1972, James: BX ⊂ BX ∗∗ boundary, characterization of
reflexivity;4 1972, Simons: H τp(B)-seq.compact and B any boundary;5 1974, de Wilde: H convex and B any boundary;6 1982, Bourgain-Talagrand: B = Ext(BX ∗), arbitrary H.
![Page 11: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/11.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
The boundary problem (Godefroy)...extremal test for compactness
Let X Banach space, B ⊂ BX ∗ boundary and denote by τp(B) thetopology defined on X by the pointwise convergence on B. Let Hbe a norm bounded and τp(B)-compact subset of X .
Is H weakly compact?
Pioneer positive known results:
1 1952, Grothendieck: X = C (K ) y B = Ext(BC(K)∗);2 1963, Rainwater: B = Ext(BX ∗), H τp(B)-seq.compact;
3 1972, James: BX ⊂ BX ∗∗ boundary, characterization ofreflexivity;
4 1972, Simons: H τp(B)-seq.compact and B any boundary;5 1974, de Wilde: H convex and B any boundary;6 1982, Bourgain-Talagrand: B = Ext(BX ∗), arbitrary H.
![Page 12: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/12.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
The boundary problem (Godefroy)...extremal test for compactness
Let X Banach space, B ⊂ BX ∗ boundary and denote by τp(B) thetopology defined on X by the pointwise convergence on B. Let Hbe a norm bounded and τp(B)-compact subset of X .
Is H weakly compact?
Pioneer positive known results:
1 1952, Grothendieck: X = C (K ) y B = Ext(BC(K)∗);2 1963, Rainwater: B = Ext(BX ∗), H τp(B)-seq.compact;3 1972, James: BX ⊂ BX ∗∗ boundary, characterization of
reflexivity;
4 1972, Simons: H τp(B)-seq.compact and B any boundary;5 1974, de Wilde: H convex and B any boundary;6 1982, Bourgain-Talagrand: B = Ext(BX ∗), arbitrary H.
![Page 13: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/13.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
The boundary problem (Godefroy)...extremal test for compactness
Let X Banach space, B ⊂ BX ∗ boundary and denote by τp(B) thetopology defined on X by the pointwise convergence on B. Let Hbe a norm bounded and τp(B)-compact subset of X .
Is H weakly compact?
Pioneer positive known results:
1 1952, Grothendieck: X = C (K ) y B = Ext(BC(K)∗);2 1963, Rainwater: B = Ext(BX ∗), H τp(B)-seq.compact;3 1972, James: BX ⊂ BX ∗∗ boundary, characterization of
reflexivity;4 1972, Simons: H τp(B)-seq.compact and B any boundary;
5 1974, de Wilde: H convex and B any boundary;6 1982, Bourgain-Talagrand: B = Ext(BX ∗), arbitrary H.
![Page 14: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/14.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
The boundary problem (Godefroy)...extremal test for compactness
Let X Banach space, B ⊂ BX ∗ boundary and denote by τp(B) thetopology defined on X by the pointwise convergence on B. Let Hbe a norm bounded and τp(B)-compact subset of X .
Is H weakly compact?
Pioneer positive known results:
1 1952, Grothendieck: X = C (K ) y B = Ext(BC(K)∗);2 1963, Rainwater: B = Ext(BX ∗), H τp(B)-seq.compact;3 1972, James: BX ⊂ BX ∗∗ boundary, characterization of
reflexivity;4 1972, Simons: H τp(B)-seq.compact and B any boundary;5 1974, de Wilde: H convex and B any boundary;
6 1982, Bourgain-Talagrand: B = Ext(BX ∗), arbitrary H.
![Page 15: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/15.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem
The boundary problem (Godefroy)...extremal test for compactness
Let X Banach space, B ⊂ BX ∗ boundary and denote by τp(B) thetopology defined on X by the pointwise convergence on B. Let Hbe a norm bounded and τp(B)-compact subset of X .
Is H weakly compact?
Pioneer positive known results:
1 1952, Grothendieck: X = C (K ) y B = Ext(BC(K)∗);2 1963, Rainwater: B = Ext(BX ∗), H τp(B)-seq.compact;3 1972, James: BX ⊂ BX ∗∗ boundary, characterization of
reflexivity;4 1972, Simons: H τp(B)-seq.compact and B any boundary;5 1974, de Wilde: H convex and B any boundary;6 1982, Bourgain-Talagrand: B = Ext(BX ∗), arbitrary H.
![Page 16: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/16.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Boundary problem for C (K )
G. Godefroy and B. C., 1998
Let K be a compact space and B ⊂ BC(K)∗ a boundary. Then asubset H of C (K ) is weakly compact if, and only if, it is normbounded and τp(B)-compact.
How can you prove something like that?
Lemma
Let K be a compact space and B ⊂ BC(K)∗ a boundary. Given asequence (fn) in C (K ) and x ∈ K , then there is µ ∈ B such that
fn(x) =∫K
fndµ for every n ∈ N.
![Page 17: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/17.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Boundary problem for C (K )
G. Godefroy and B. C., 1998
Let K be a compact space and B ⊂ BC(K)∗ a boundary. Then asubset H of C (K ) is weakly compact if, and only if, it is normbounded and τp(B)-compact.
How can you prove something like that?
Lemma
Let K be a compact space and B ⊂ BC(K)∗ a boundary. Given asequence (fn) in C (K ) and x ∈ K , then there is µ ∈ B such that
fn(x) =∫K
fndµ for every n ∈ N.
![Page 18: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/18.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Boundary problem for C (K )
G. Godefroy and B. C., 1998
Let K be a compact space and B ⊂ BC(K)∗ a boundary. Then asubset H of C (K ) is weakly compact if, and only if, it is normbounded and τp(B)-compact.
How can you prove something like that?
Lemma
Let K be a compact space and B ⊂ BC(K)∗ a boundary. Given asequence (fn) in C (K ) and x ∈ K , then there is µ ∈ B such that
fn(x) =∫K
fndµ for every n ∈ N.
![Page 19: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/19.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Boundary problem for C (K )
LemmaLet K be a compact space and B ⊂ BC(K)∗ a boundary. Given a sequence (fn) in C(K) and x ∈K , then there is
µ ∈ B such that fn(x) =∫K fndµ for every n ∈ N.
Proof.-
X g(t) := 1−∑∞n=1
12n
|fn(t)−fn(x)|1+|fn(t)−fn(x)| , t ∈ K , 0≤ g ≤ 1.
X Then F =⋂
∞n=1y ∈ K ; fn(y) = fn(x)= y ∈ K : g(y) = 1 = ‖g‖∞.
X ∃ µ ∈ B such that 1 =∫K gdµ ≤
∫K gd |µ| ≤ |µ|(K)≤ ‖µ‖= 1.
X In different words,
0 = |µ|(K)−∫K
gd |µ|=∫K
(1−g)d |µ|.
X Since 1−g ≥ 0 we obtain 0 = |µ|(y ∈ K : 1−g(y) > 0) = |µ|(K \F )
X Then for every n ∈ N,∫K
fndµ =∫F
fndµ =∫F
fn(x)dµ = fn(x)
because µ is a probability itself.
![Page 20: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/20.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Boundary problem for C (K )
LemmaLet K be a compact space and B ⊂ BC(K)∗ a boundary. Given a sequence (fn) in C(K) and x ∈K , then there is
µ ∈ B such that fn(x) =∫K fndµ for every n ∈ N.
Proof.-
X g(t) := 1−∑∞n=1
12n
|fn(t)−fn(x)|1+|fn(t)−fn(x)| , t ∈ K , 0≤ g ≤ 1.
X Then F =⋂
∞n=1y ∈ K ; fn(y) = fn(x)= y ∈ K : g(y) = 1 = ‖g‖∞.
X ∃ µ ∈ B such that 1 =∫K gdµ ≤
∫K gd |µ| ≤ |µ|(K)≤ ‖µ‖= 1.
X In different words,
0 = |µ|(K)−∫K
gd |µ|=∫K
(1−g)d |µ|.
X Since 1−g ≥ 0 we obtain 0 = |µ|(y ∈ K : 1−g(y) > 0) = |µ|(K \F )
X Then for every n ∈ N,∫K
fndµ =∫F
fndµ =∫F
fn(x)dµ = fn(x)
because µ is a probability itself.
![Page 21: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/21.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Boundary problem for C (K )
LemmaLet K be a compact space and B ⊂ BC(K)∗ a boundary. Given a sequence (fn) in C(K) and x ∈K , then there is
µ ∈ B such that fn(x) =∫K fndµ for every n ∈ N.
Proof.-
X g(t) := 1−∑∞n=1
12n
|fn(t)−fn(x)|1+|fn(t)−fn(x)| , t ∈ K , 0≤ g ≤ 1.
X Then F =⋂
∞n=1y ∈ K ; fn(y) = fn(x)= y ∈ K : g(y) = 1 = ‖g‖∞.
X ∃ µ ∈ B such that 1 =∫K gdµ ≤
∫K gd |µ| ≤ |µ|(K)≤ ‖µ‖= 1.
X In different words,
0 = |µ|(K)−∫K
gd |µ|=∫K
(1−g)d |µ|.
X Since 1−g ≥ 0 we obtain 0 = |µ|(y ∈ K : 1−g(y) > 0) = |µ|(K \F )
X Then for every n ∈ N,∫K
fndµ =∫F
fndµ =∫F
fn(x)dµ = fn(x)
because µ is a probability itself.
![Page 22: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/22.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Boundary problem for C (K )
LemmaLet K be a compact space and B ⊂ BC(K)∗ a boundary. Given a sequence (fn) in C(K) and x ∈K , then there is
µ ∈ B such that fn(x) =∫K fndµ for every n ∈ N.
Proof.-
X g(t) := 1−∑∞n=1
12n
|fn(t)−fn(x)|1+|fn(t)−fn(x)| , t ∈ K , 0≤ g ≤ 1.
X Then F =⋂
∞n=1y ∈ K ; fn(y) = fn(x)= y ∈ K : g(y) = 1 = ‖g‖∞.
X ∃ µ ∈ B such that 1 =∫K gdµ ≤
∫K gd |µ| ≤ |µ|(K)≤ ‖µ‖= 1.
X In different words,
0 = |µ|(K)−∫K
gd |µ|=∫K
(1−g)d |µ|.
X Since 1−g ≥ 0 we obtain 0 = |µ|(y ∈ K : 1−g(y) > 0) = |µ|(K \F )
X Then for every n ∈ N,∫K
fndµ =∫F
fndµ =∫F
fn(x)dµ = fn(x)
because µ is a probability itself.
![Page 23: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/23.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Boundary problem for C (K )
LemmaLet K be a compact space and B ⊂ BC(K)∗ a boundary. Given a sequence (fn) in C(K) and x ∈K , then there is
µ ∈ B such that fn(x) =∫K fndµ for every n ∈ N.
Proof.-
X g(t) := 1−∑∞n=1
12n
|fn(t)−fn(x)|1+|fn(t)−fn(x)| , t ∈ K , 0≤ g ≤ 1.
X Then F =⋂
∞n=1y ∈ K ; fn(y) = fn(x)= y ∈ K : g(y) = 1 = ‖g‖∞.
X ∃ µ ∈ B such that 1 =∫K gdµ ≤
∫K gd |µ| ≤ |µ|(K)≤ ‖µ‖= 1.
X In different words,
0 = |µ|(K)−∫K
gd |µ|=∫K
(1−g)d |µ|.
X Since 1−g ≥ 0 we obtain 0 = |µ|(y ∈ K : 1−g(y) > 0) = |µ|(K \F )
X Then for every n ∈ N,∫K
fndµ =∫F
fndµ =∫F
fn(x)dµ = fn(x)
because µ is a probability itself.
![Page 24: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/24.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Boundary problem for C (K )
LemmaLet K be a compact space and B ⊂ BC(K)∗ a boundary. Given a sequence (fn) in C(K) and x ∈K , then there is
µ ∈ B such that fn(x) =∫K fndµ for every n ∈ N.
Proof.-
X g(t) := 1−∑∞n=1
12n
|fn(t)−fn(x)|1+|fn(t)−fn(x)| , t ∈ K , 0≤ g ≤ 1.
X Then F =⋂
∞n=1y ∈ K ; fn(y) = fn(x)= y ∈ K : g(y) = 1 = ‖g‖∞.
X ∃ µ ∈ B such that 1 =∫K gdµ ≤
∫K gd |µ| ≤ |µ|(K)≤ ‖µ‖= 1.
X In different words,
0 = |µ|(K)−∫K
gd |µ|=∫K
(1−g)d |µ|.
X Since 1−g ≥ 0 we obtain 0 = |µ|(y ∈ K : 1−g(y) > 0) = |µ|(K \F )
X Then for every n ∈ N,∫K
fndµ =∫F
fndµ =∫F
fn(x)dµ = fn(x)
because µ is a probability itself.
![Page 25: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/25.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Boundary problem for C (K )
LemmaLet K be a compact space and B ⊂ BC(K)∗ a boundary. Given a sequence (fn) in C(K) and x ∈K , then there is
µ ∈ B such that fn(x) =∫K fndµ for every n ∈ N.
Proof.-
X g(t) := 1−∑∞n=1
12n
|fn(t)−fn(x)|1+|fn(t)−fn(x)| , t ∈ K , 0≤ g ≤ 1.
X Then F =⋂
∞n=1y ∈ K ; fn(y) = fn(x)= y ∈ K : g(y) = 1 = ‖g‖∞.
X ∃ µ ∈ B such that 1 =∫K gdµ ≤
∫K gd |µ| ≤ |µ|(K)≤ ‖µ‖= 1.
X In different words,
0 = |µ|(K)−∫K
gd |µ|=∫K
(1−g)d |µ|.
X Since 1−g ≥ 0 we obtain 0 = |µ|(y ∈ K : 1−g(y) > 0) = |µ|(K \F )
X Then for every n ∈ N,∫K
fndµ =∫F
fndµ =∫F
fn(x)dµ = fn(x)
because µ is a probability itself.
![Page 26: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/26.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem for general Banach spaces X
Key point...de Wilde’s result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is convex andτp(B)-compact then H is weakly compact.
if I want to solve
Boundary problem
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is normbounded and τp(B)-compact then H is weakly compact.
I wish I had a tool...
...Krein-Smulyan type result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is norm
bounded and τp(B)-compact, then co(H)τp(B)
is τp(B)-compact.
X Bad news: τp(B) is not compatible with 〈X ,X ∗〉.X Good news: we can overcome the difficulties for many Banach spaces.
![Page 27: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/27.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem for general Banach spaces X
Key point...de Wilde’s result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is convex andτp(B)-compact then H is weakly compact.
if I want to solve
Boundary problem
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is normbounded and τp(B)-compact then H is weakly compact.
I wish I had a tool...
...Krein-Smulyan type result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is norm
bounded and τp(B)-compact, then co(H)τp(B)
is τp(B)-compact.
X Bad news: τp(B) is not compatible with 〈X ,X ∗〉.X Good news: we can overcome the difficulties for many Banach spaces.
![Page 28: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/28.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem for general Banach spaces X
Key point...de Wilde’s result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is convex andτp(B)-compact then H is weakly compact.
if I want to solve
Boundary problem
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is normbounded and τp(B)-compact then H is weakly compact.
I wish I had a tool...
...Krein-Smulyan type result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is norm
bounded and τp(B)-compact, then co(H)τp(B)
is τp(B)-compact.
X Bad news: τp(B) is not compatible with 〈X ,X ∗〉.X Good news: we can overcome the difficulties for many Banach spaces.
![Page 29: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/29.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem for general Banach spaces X
Key point...de Wilde’s result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is convex andτp(B)-compact then H is weakly compact.
if I want to solve
Boundary problem
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is normbounded and τp(B)-compact then H is weakly compact.
I wish I had a tool...
...Krein-Smulyan type result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is norm
bounded and τp(B)-compact, then co(H)τp(B)
is τp(B)-compact.
X Bad news: τp(B) is not compatible with 〈X ,X ∗〉.X Good news: we can overcome the difficulties for many Banach spaces.
![Page 30: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/30.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem for general Banach spaces X
Key point...de Wilde’s result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is convex andτp(B)-compact then H is weakly compact.
if I want to solve
Boundary problem
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is normbounded and τp(B)-compact then H is weakly compact.
I wish I had a tool...
...Krein-Smulyan type result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is norm
bounded and τp(B)-compact, then co(H)τp(B)
is τp(B)-compact.
X Bad news: τp(B) is not compatible with 〈X ,X ∗〉.
X Good news: we can overcome the difficulties for many Banach spaces.
![Page 31: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/31.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The boundary problem for general Banach spaces X
Key point...de Wilde’s result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is convex andτp(B)-compact then H is weakly compact.
if I want to solve
Boundary problem
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is normbounded and τp(B)-compact then H is weakly compact.
I wish I had a tool...
...Krein-Smulyan type result
Let X be a Banach space and B a boundary for BX ∗ . If H ⊂ X is norm
bounded and τp(B)-compact, then co(H)τp(B)
is τp(B)-compact.
X Bad news: τp(B) is not compatible with 〈X ,X ∗〉.X Good news: we can overcome the difficulties for many Banach spaces.
![Page 32: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/32.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Looking for inspiration. . .
![Page 33: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/33.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Looking for inspiration. . .
![Page 34: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/34.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Looking for inspiration. . .
![Page 35: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/35.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-compact subset H of X its τp(B)-closed convex
hull co(H)τp(B)
is τp(B)-compact.
Proof.- Fix µ a Radon probability on (H,τp(B)), find a barycenter for µ?
find xµ ∈ X with x∗(xµ ) =∫H
x∗|Hdµ, for every x∗ ∈ X ∗?
X Difficulty: x∗|H is measurable only for x∗ ∈ B;
X since τp(B) = τp(co(B)), we can assume B convex;
X if B is convex and 1-norming then Hahn-Banach implies Bw ∗
= BX ∗ .
X B|H := x∗|H : x∗ ∈ B ⊂ C(H,τp(B))
and BX ∗ |H = B|Hτp(H)
;
X if B|H does not have independent sequences (Rosenthal) then B|Hτp(H)
ismade up of µ-measurable functions for each µ;
X indeed, B|H as above has Bourgain property with respect to µ.
(b∗n)n in B is independent on H if there are s < t such that(⋂n∈P
w ∈ H : b∗n(w) < s
)∩
( ⋂n∈Q
w ∈ H : b∗n(w) > t
)
for every disjoint finite sets P,Q ⊂ N.
![Page 36: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/36.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-compact subset H of X its τp(B)-closed convex
hull co(H)τp(B)
is τp(B)-compact.
Proof.- Fix µ a Radon probability on (H,τp(B)), find a barycenter for µ?
find xµ ∈ X with x∗(xµ ) =∫H
x∗|Hdµ, for every x∗ ∈ X ∗?
X Difficulty: x∗|H is measurable only for x∗ ∈ B;
X since τp(B) = τp(co(B)), we can assume B convex;
X if B is convex and 1-norming then Hahn-Banach implies Bw ∗
= BX ∗ .
X B|H := x∗|H : x∗ ∈ B ⊂ C(H,τp(B)) and BX ∗ |H = B|Hτp(H)
;
X if B|H does not have independent sequences (Rosenthal) then B|Hτp(H)
ismade up of µ-measurable functions for each µ;
X indeed, B|H as above has Bourgain property with respect to µ.
(b∗n)n in B is independent on H if there are s < t such that(⋂n∈P
w ∈ H : b∗n(w) < s
)∩
( ⋂n∈Q
w ∈ H : b∗n(w) > t
)
for every disjoint finite sets P,Q ⊂ N.
![Page 37: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/37.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-compact subset H of X its τp(B)-closed convex
hull co(H)τp(B)
is τp(B)-compact.
Proof.- Fix µ a Radon probability on (H,τp(B)), find a barycenter for µ?
find xµ ∈ X with x∗(xµ ) =∫H
x∗|Hdµ, for every x∗ ∈ X ∗?
X Difficulty: x∗|H is measurable only for x∗ ∈ B;
X since τp(B) = τp(co(B)), we can assume B convex;
X if B is convex and 1-norming then Hahn-Banach implies Bw ∗
= BX ∗ .
X B|H := x∗|H : x∗ ∈ B ⊂ C(H,τp(B)) and BX ∗ |H = B|Hτp(H)
;
X if B|H does not have independent sequences (Rosenthal) then B|Hτp(H)
ismade up of µ-measurable functions for each µ;
X indeed, B|H as above has Bourgain property with respect to µ.
(b∗n)n in B is independent on H if there are s < t such that(⋂n∈P
w ∈ H : b∗n(w) < s
)∩
( ⋂n∈Q
w ∈ H : b∗n(w) > t
)
for every disjoint finite sets P,Q ⊂ N.
![Page 38: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/38.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-compact subset H of X its τp(B)-closed convex
hull co(H)τp(B)
is τp(B)-compact.
Proof.- Fix µ a Radon probability on (H,τp(B)), find a barycenter for µ?
find xµ ∈ X with x∗(xµ ) =∫H
x∗|Hdµ, for every x∗ ∈ X ∗?
X Difficulty: x∗|H is measurable only for x∗ ∈ B;
X since τp(B) = τp(co(B)), we can assume B convex;
X if B is convex and 1-norming then Hahn-Banach implies Bw ∗
= BX ∗ .
X B|H := x∗|H : x∗ ∈ B ⊂ C(H,τp(B)) and BX ∗ |H = B|Hτp(H)
;
X if B|H does not have independent sequences (Rosenthal) then B|Hτp(H)
ismade up of µ-measurable functions for each µ;
X indeed, B|H as above has Bourgain property with respect to µ.
(b∗n)n in B is independent on H if there are s < t such that(⋂n∈P
w ∈ H : b∗n(w) < s
)∩
( ⋂n∈Q
w ∈ H : b∗n(w) > t
)
for every disjoint finite sets P,Q ⊂ N.
![Page 39: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/39.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-compact subset H of X its τp(B)-closed convex
hull co(H)τp(B)
is τp(B)-compact.
Proof.- Fix µ a Radon probability on (H,τp(B)), find a barycenter for µ?
find xµ ∈ X with x∗(xµ ) =∫H
x∗|Hdµ, for every x∗ ∈ X ∗?
X Difficulty: x∗|H is measurable only for x∗ ∈ B;
X since τp(B) = τp(co(B)), we can assume B convex;
X if B is convex and 1-norming then Hahn-Banach implies Bw ∗
= BX ∗ .
X B|H := x∗|H : x∗ ∈ B ⊂ C(H,τp(B)) and BX ∗ |H = B|Hτp(H)
;
X if B|H does not have independent sequences (Rosenthal) then B|Hτp(H)
ismade up of µ-measurable functions for each µ;
X indeed, B|H as above has Bourgain property with respect to µ.
(b∗n)n in B is independent on H if there are s < t such that(⋂n∈P
w ∈ H : b∗n(w) < s
)∩
( ⋂n∈Q
w ∈ H : b∗n(w) > t
)
for every disjoint finite sets P,Q ⊂ N.
![Page 40: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/40.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-compact subset H of X its τp(B)-closed convex
hull co(H)τp(B)
is τp(B)-compact.
Proof.- Fix µ a Radon probability on (H,τp(B)), find a barycenter for µ?
find xµ ∈ X with x∗(xµ ) =∫H
x∗|Hdµ, for every x∗ ∈ X ∗?
X Difficulty: x∗|H is measurable only for x∗ ∈ B;
X since τp(B) = τp(co(B)), we can assume B convex;
X if B is convex and 1-norming then Hahn-Banach implies Bw ∗
= BX ∗ .
X B|H := x∗|H : x∗ ∈ B ⊂ C(H,τp(B)) and BX ∗ |H = B|Hτp(H)
;
X if B|H does not have independent sequences (Rosenthal) then B|Hτp(H)
ismade up of µ-measurable functions for each µ;
X indeed, B|H as above has Bourgain property with respect to µ.
(b∗n)n in B is independent on H if there are s < t such that(⋂n∈P
w ∈ H : b∗n(w) < s
)∩
( ⋂n∈Q
w ∈ H : b∗n(w) > t
)
for every disjoint finite sets P,Q ⊂ N.
![Page 41: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/41.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-compact subset H of X its τp(B)-closed convex
hull co(H)τp(B)
is τp(B)-compact.
Proof.- Fix µ a Radon probability on (H,τp(B)), find a barycenter for µ?
find xµ ∈ X with x∗(xµ ) =∫H
x∗|Hdµ, for every x∗ ∈ X ∗?
X Difficulty: x∗|H is measurable only for x∗ ∈ B;
X since τp(B) = τp(co(B)), we can assume B convex;
X if B is convex and 1-norming then Hahn-Banach implies Bw ∗
= BX ∗ .
X B|H := x∗|H : x∗ ∈ B ⊂ C(H,τp(B)) and BX ∗ |H = B|Hτp(H)
;
X if B|H does not have independent sequences (Rosenthal) then B|Hτp(H)
ismade up of µ-measurable functions for each µ;
X indeed, B|H as above has Bourgain property with respect to µ.
(b∗n)n in B is independent on H if there are s < t such that(⋂n∈P
w ∈ H : b∗n(w) < s
)∩
( ⋂n∈Q
w ∈ H : b∗n(w) > t
)
for every disjoint finite sets P,Q ⊂ N.
![Page 42: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/42.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-compact subset H of X its τp(B)-closed convex
hull co(H)τp(B)
is τp(B)-compact.
Proof.- Fix µ a Radon probability on (H,τp(B)), find a barycenter for µ?
find xµ ∈ X with x∗(xµ ) =∫H
x∗|Hdµ, for every x∗ ∈ X ∗?
X Difficulty: x∗|H is measurable only for x∗ ∈ B;
X since τp(B) = τp(co(B)), we can assume B convex;
X if B is convex and 1-norming then Hahn-Banach implies Bw ∗
= BX ∗ .
X B|H := x∗|H : x∗ ∈ B ⊂ C(H,τp(B)) and BX ∗ |H = B|Hτp(H)
;
X if B|H does not have independent sequences (Rosenthal) then B|Hτp(H)
ismade up of µ-measurable functions for each µ;
X indeed, B|H as above has Bourgain property with respect to µ.
(b∗n)n in B is independent on H if there are s < t such that(⋂n∈P
w ∈ H : b∗n(w) < s
)∩
( ⋂n∈Q
w ∈ H : b∗n(w) > t
)
for every disjoint finite sets P,Q ⊂ N.
![Page 43: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/43.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Bourgain property. . . a bit of history
Definition
We say that a family F ⊂ RΩ has Bourgainproperty if for every ε > 0 and every A ∈Σwith µ(A) > 0 there are B1, . . . ,Bn ⊂ A, Bi ∈Σ,with µ(Bi ) > 0 such that for every f ∈F
inf1≤i≤n
| · |diam(f (Bi )) < ε.
The property of Bourgain
The notion wasn’t published by Bourgain.
It appears in a paper by [RS85] and refersto handwritten notes by Bourgain.
![Page 44: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/44.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Bourgain property. . . a bit of history
Definition
We say that a family F ⊂ RΩ has Bourgainproperty if for every ε > 0 and every A ∈Σwith µ(A) > 0 there are B1, . . . ,Bn ⊂ A, Bi ∈Σ,with µ(Bi ) > 0 such that for every f ∈F
inf1≤i≤n
| · |diam(f (Bi )) < ε.
The property of Bourgain
The notion wasn’t published by Bourgain.
It appears in a paper by [RS85] and refersto handwritten notes by Bourgain.
![Page 45: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/45.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Bourgain property. . . a bit of history
Definition
We say that a family F ⊂ RΩ has Bourgainproperty if for every ε > 0 and every A ∈Σwith µ(A) > 0 there are B1, . . . ,Bn ⊂ A, Bi ∈Σ,with µ(Bi ) > 0 such that for every f ∈F
inf1≤i≤n
| · |diam(f (Bi )) < ε.
The property of Bourgain
The notion wasn’t published by Bourgain.
It appears in a paper by [RS85] and refersto handwritten notes by Bourgain.
![Page 46: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/46.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Remarkable facts about Bourgain property
Bourgain Property
We say that a family F ⊂ RΩ has Bourgain property if for every ε > 0 and every A ∈Σ with µ(A) > 0 there areB1 , . . . ,Bn ⊂ A, Bi ∈Σ, with µ(Bi ) > 0 such that for every f ∈F
inf1≤i≤n
| · |diam(f (Bi )) < ε.
Properties
If F = f , TFAE:
(i) (Bourgain property) For every ε > 0 and every A ∈Σ with µ(A) > 0there is B ∈Σ, B ⊂ A with µ(B) > 0 and | · |diam f (B) < ε.
(ii) f is measurable.
If F has Bourgain property, then F is made up of measurable functions.
F has Bourgain property ⇒ Fτp(Ω)
has too.
F has Bourgain property and f ∈Fτp(Ω)
, then there is a sequence (fn) inF that converges to f , µ-almost everywhere.
![Page 47: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/47.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Remarkable facts about Bourgain property
Bourgain Property
We say that a family F ⊂ RΩ has Bourgain property if for every ε > 0 and every A ∈Σ with µ(A) > 0 there areB1 , . . . ,Bn ⊂ A, Bi ∈Σ, with µ(Bi ) > 0 such that for every f ∈F
inf1≤i≤n
| · |diam(f (Bi )) < ε.
Properties
If F = f , TFAE:
(i) (Bourgain property) For every ε > 0 and every A ∈Σ with µ(A) > 0there is B ∈Σ, B ⊂ A with µ(B) > 0 and | · |diam f (B) < ε.
(ii) f is measurable.
If F has Bourgain property, then F is made up of measurable functions.
F has Bourgain property ⇒ Fτp(Ω)
has too.
F has Bourgain property and f ∈Fτp(Ω)
, then there is a sequence (fn) inF that converges to f , µ-almost everywhere.
![Page 48: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/48.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Remarkable facts about Bourgain property
Bourgain Property
We say that a family F ⊂ RΩ has Bourgain property if for every ε > 0 and every A ∈Σ with µ(A) > 0 there areB1 , . . . ,Bn ⊂ A, Bi ∈Σ, with µ(Bi ) > 0 such that for every f ∈F
inf1≤i≤n
| · |diam(f (Bi )) < ε.
Properties
If F = f , TFAE:
(i) (Bourgain property) For every ε > 0 and every A ∈Σ with µ(A) > 0there is B ∈Σ, B ⊂ A with µ(B) > 0 and | · |diam f (B) < ε.
(ii) f is measurable.
If F has Bourgain property, then F is made up of measurable functions.
F has Bourgain property ⇒ Fτp(Ω)
has too.
F has Bourgain property and f ∈Fτp(Ω)
, then there is a sequence (fn) inF that converges to f , µ-almost everywhere.
![Page 49: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/49.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Remarkable facts about Bourgain property
Bourgain Property
We say that a family F ⊂ RΩ has Bourgain property if for every ε > 0 and every A ∈Σ with µ(A) > 0 there areB1 , . . . ,Bn ⊂ A, Bi ∈Σ, with µ(Bi ) > 0 such that for every f ∈F
inf1≤i≤n
| · |diam(f (Bi )) < ε.
Properties
If F = f , TFAE:
(i) (Bourgain property) For every ε > 0 and every A ∈Σ with µ(A) > 0there is B ∈Σ, B ⊂ A with µ(B) > 0 and | · |diam f (B) < ε.
(ii) f is measurable.
If F has Bourgain property, then F is made up of measurable functions.
F has Bourgain property ⇒ Fτp(Ω)
has too.
F has Bourgain property and f ∈Fτp(Ω)
, then there is a sequence (fn) inF that converges to f , µ-almost everywhere.
![Page 50: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/50.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
. . . back to Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact.
X if B|H does not have independent sequences (Rosenthal), then B|H hasBourgain property with respect to µ.
X BX ∗ |H = B|Hτp(H)
has Bourgain property;
X we can write Tµ (x∗) =∫H x∗|Hdµ and Tµ ∈ X ∗∗;
X given any A⊂ BX ∗ if x∗|H ∈ A|Hτp(H)
, then there is a sequence (x∗n |H) inA|H that converges to x∗|H , µ-almost everywhere.
X then for each A⊂ BX ∗ , Tµ (Aw ∗
)⊂ Tµ (A);
X Grothendieck completeness⇒ Tµ = xµ ∈ X barycenter. . . goes onsmoothly.
X if B|H has an independent sequence on H ⇒ βN ⊂ (BX ∗ ,w∗).
![Page 51: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/51.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
. . . back to Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact.
X if B|H does not have independent sequences (Rosenthal), then B|H hasBourgain property with respect to µ.
X BX ∗ |H = B|Hτp(H)
has Bourgain property;
X we can write Tµ (x∗) =∫H x∗|Hdµ and Tµ ∈ X ∗∗;
X given any A⊂ BX ∗ if x∗|H ∈ A|Hτp(H)
, then there is a sequence (x∗n |H) inA|H that converges to x∗|H , µ-almost everywhere.
X then for each A⊂ BX ∗ , Tµ (Aw ∗
)⊂ Tµ (A);
X Grothendieck completeness⇒ Tµ = xµ ∈ X barycenter. . . goes onsmoothly.
X if B|H has an independent sequence on H ⇒ βN ⊂ (BX ∗ ,w∗).
![Page 52: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/52.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
. . . back to Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact.
X if B|H does not have independent sequences (Rosenthal), then B|H hasBourgain property with respect to µ.
X BX ∗ |H = B|Hτp(H)
has Bourgain property;
X we can write Tµ (x∗) =∫H x∗|Hdµ and Tµ ∈ X ∗∗;
X given any A⊂ BX ∗ if x∗|H ∈ A|Hτp(H)
, then there is a sequence (x∗n |H) inA|H that converges to x∗|H , µ-almost everywhere.
X then for each A⊂ BX ∗ , Tµ (Aw ∗
)⊂ Tµ (A);
X Grothendieck completeness⇒ Tµ = xµ ∈ X barycenter. . . goes onsmoothly.
X if B|H has an independent sequence on H ⇒ βN ⊂ (BX ∗ ,w∗).
![Page 53: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/53.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
. . . back to Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact.
X if B|H does not have independent sequences (Rosenthal), then B|H hasBourgain property with respect to µ.
X BX ∗ |H = B|Hτp(H)
has Bourgain property;
X we can write Tµ (x∗) =∫H x∗|Hdµ and Tµ ∈ X ∗∗;
X given any A⊂ BX ∗ if x∗|H ∈ A|Hτp(H)
, then there is a sequence (x∗n |H) inA|H that converges to x∗|H , µ-almost everywhere.
X then for each A⊂ BX ∗ , Tµ (Aw ∗
)⊂ Tµ (A);
X Grothendieck completeness⇒ Tµ = xµ ∈ X barycenter. . . goes onsmoothly.
X if B|H has an independent sequence on H ⇒ βN ⊂ (BX ∗ ,w∗).
![Page 54: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/54.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
. . . back to Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact.
X if B|H does not have independent sequences (Rosenthal), then B|H hasBourgain property with respect to µ.
X BX ∗ |H = B|Hτp(H)
has Bourgain property;
X we can write Tµ (x∗) =∫H x∗|Hdµ and Tµ ∈ X ∗∗;
X given any A⊂ BX ∗ if x∗|H ∈ A|Hτp(H)
, then there is a sequence (x∗n |H) inA|H that converges to x∗|H , µ-almost everywhere.
X then for each A⊂ BX ∗ , Tµ (Aw ∗
)⊂ Tµ (A);
X Grothendieck completeness⇒ Tµ = xµ ∈ X barycenter. . . goes onsmoothly.
X if B|H has an independent sequence on H ⇒ βN ⊂ (BX ∗ ,w∗).
![Page 55: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/55.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
. . . back to Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact.
X if B|H does not have independent sequences (Rosenthal), then B|H hasBourgain property with respect to µ.
X BX ∗ |H = B|Hτp(H)
has Bourgain property;
X we can write Tµ (x∗) =∫H x∗|Hdµ and Tµ ∈ X ∗∗;
X given any A⊂ BX ∗ if x∗|H ∈ A|Hτp(H)
, then there is a sequence (x∗n |H) inA|H that converges to x∗|H , µ-almost everywhere.
X then for each A⊂ BX ∗ , Tµ (Aw ∗
)⊂ Tµ (A);
X Grothendieck completeness⇒ Tµ = xµ ∈ X barycenter. . . goes onsmoothly.
X if B|H has an independent sequence on H ⇒ βN ⊂ (BX ∗ ,w∗).
![Page 56: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/56.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
. . . back to Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact.
X if B|H does not have independent sequences (Rosenthal), then B|H hasBourgain property with respect to µ.
X BX ∗ |H = B|Hτp(H)
has Bourgain property;
X we can write Tµ (x∗) =∫H x∗|Hdµ and Tµ ∈ X ∗∗;
X given any A⊂ BX ∗ if x∗|H ∈ A|Hτp(H)
, then there is a sequence (x∗n |H) inA|H that converges to x∗|H , µ-almost everywhere.
X then for each A⊂ BX ∗ , Tµ (Aw ∗
)⊂ Tµ (A);
X Grothendieck completeness⇒ Tµ = xµ ∈ X barycenter. . . goes onsmoothly.
X if B|H has an independent sequence on H ⇒ βN ⊂ (BX ∗ ,w∗).
![Page 57: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/57.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
. . . back to Krein-Smulyan type result
Wish. . .
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact.
X if B|H does not have independent sequences (Rosenthal), then B|H hasBourgain property with respect to µ.
X BX ∗ |H = B|Hτp(H)
has Bourgain property;
X we can write Tµ (x∗) =∫H x∗|Hdµ and Tµ ∈ X ∗∗;
X given any A⊂ BX ∗ if x∗|H ∈ A|Hτp(H)
, then there is a sequence (x∗n |H) inA|H that converges to x∗|H , µ-almost everywhere.
X then for each A⊂ BX ∗ , Tµ (Aw ∗
)⊂ Tµ (A);
X Grothendieck completeness⇒ Tµ = xµ ∈ X barycenter. . . goes onsmoothly.
X if B|H has an independent sequence on H ⇒ βN ⊂ (BX ∗ ,w∗).
![Page 58: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/58.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
. . . back to Krein-Smulyan type result
Theorem: Manjabacas, Vera and B.C., 1997
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact, assuming βN 6⊂ (BX ∗ ,w∗).
X if B|H does not have independent sequences (Rosenthal), then B|H hasBourgain property with respect to µ.
X BX ∗ |H = B|Hτp(H)
has Bourgain property;
X we can write Tµ (x∗) =∫H x∗|Hdµ and Tµ ∈ X ∗∗;
X given any A⊂ BX ∗ if x∗|H ∈ A|Hτp(H)
, then there is a sequence (x∗n |H) inA|H that converges to x∗|H , µ-almost everywhere.
X then for each A⊂ BX ∗ , Tµ (Aw ∗
)⊂ Tµ (A);
X Grothendieck completeness⇒ Tµ = xµ ∈ X barycenter. . . goes onsmoothly.
X if B|H has an independent sequence on H ⇒ βN ⊂ (BX ∗ ,w∗).
![Page 59: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/59.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
. . . back to Krein-Smulyan type result
Theorem: Manjabacas, Vera and B.C., 1997
Take X Banach space and B ⊂ BX ∗ 1-norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact, assuming `1(c) 6⊂ X .
X if B|H does not have independent sequences (Rosenthal), then B|H hasBourgain property with respect to µ.
X BX ∗ |H = B|Hτp(H)
has Bourgain property;
X we can write Tµ (x∗) =∫H x∗|Hdµ and Tµ ∈ X ∗∗;
X given any A⊂ BX ∗ if x∗|H ∈ A|Hτp(H)
, then there is a sequence (x∗n |H) inA|H that converges to x∗|H , µ-almost everywhere.
X then for each A⊂ BX ∗ , Tµ (Aw ∗
)⊂ Tµ (A);
X Grothendieck completeness⇒ Tµ = xµ ∈ X barycenter. . . goes onsmoothly.
X if B|H has an independent sequence on H ⇒ βN ⊂ (BX ∗ ,w∗).
![Page 60: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/60.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
What we know about the boundary problem for XTheorem: Manjabacas, Vera and B.C., 1997
Take X Banach space and B ⊂ BX ∗ norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact, assuming `1(c) 6⊂ X .
Corollary: Manjabacas, Vera and B.C., 1997
Let X be a Banach space such that `1(c) 6⊂ X and B any boundary for BX ∗ . IfH ⊂ X is norm bounded and τp(B)-compact then H is weakly compact.
Is the Theorem cheap?
A. S. Granero 2006
Take X Banach space. TFAE:
1 For every B ⊂ BX ∗ norming and every norm bounded τp(B)-rel. compact
subset H of X its τp(B)-closed convex hull co(H)τp(B)
is τp(B)-compact;
2 `1(c) 6⊂ X .
Is the Corollary cheap?
![Page 61: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/61.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
What we know about the boundary problem for XTheorem: Manjabacas, Vera and B.C., 1997
Take X Banach space and B ⊂ BX ∗ norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact, assuming `1(c) 6⊂ X .
Corollary: Manjabacas, Vera and B.C., 1997
Let X be a Banach space such that `1(c) 6⊂ X and B any boundary for BX ∗ . IfH ⊂ X is norm bounded and τp(B)-compact then H is weakly compact.
Is the Theorem cheap?
A. S. Granero 2006
Take X Banach space. TFAE:
1 For every B ⊂ BX ∗ norming and every norm bounded τp(B)-rel. compact
subset H of X its τp(B)-closed convex hull co(H)τp(B)
is τp(B)-compact;
2 `1(c) 6⊂ X .
Is the Corollary cheap?
![Page 62: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/62.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
What we know about the boundary problem for XTheorem: Manjabacas, Vera and B.C., 1997
Take X Banach space and B ⊂ BX ∗ norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact, assuming `1(c) 6⊂ X .
Corollary: Manjabacas, Vera and B.C., 1997
Let X be a Banach space such that `1(c) 6⊂ X and B any boundary for BX ∗ . IfH ⊂ X is norm bounded and τp(B)-compact then H is weakly compact.
Is the Theorem cheap?
A. S. Granero 2006
Take X Banach space. TFAE:
1 For every B ⊂ BX ∗ norming and every norm bounded τp(B)-rel. compact
subset H of X its τp(B)-closed convex hull co(H)τp(B)
is τp(B)-compact;
2 `1(c) 6⊂ X .
Is the Corollary cheap?
![Page 63: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/63.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
What we know about the boundary problem for XTheorem: Manjabacas, Vera and B.C., 1997
Take X Banach space and B ⊂ BX ∗ norming (i.e. ‖x‖= supx∗(x) : x∗ ∈ B). Forevery norm bounded τp(B)-relatively compact subset H of X its τp(B)-closed
convex hull co(H)τp(B)
is τp(B)-compact, assuming `1(c) 6⊂ X .
Corollary: Manjabacas, Vera and B.C., 1997
Let X be a Banach space such that `1(c) 6⊂ X and B any boundary for BX ∗ . IfH ⊂ X is norm bounded and τp(B)-compact then H is weakly compact.
Is the Theorem cheap?
A. S. Granero 2006
Take X Banach space. TFAE:
1 For every B ⊂ BX ∗ norming and every norm bounded τp(B)-rel. compact
subset H of X its τp(B)-closed convex hull co(H)τp(B)
is τp(B)-compact;
2 `1(c) 6⊂ X .
Is the Corollary cheap?
![Page 64: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/64.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
![Page 65: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/65.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Bourgain property & Birkhoff integrability
1 Given H ⊂ X τp(B) compact and µ Radon probability wehave studied (Pettis) integrability of id : H → X usingBourgain property of
Zid = x∗ id : x∗ ∈ BX ∗ ⊂ RH .
2 In general if (Ω,Σ,µ) is a complete probability space andf : Ω → X is bounded and such that
Zf = x∗ f : x∗ ∈ BX ∗ ⊂ RΩ
has Bourgain property what can you say about f ?
Using techniques of Pettis integration the known answer is: f isPettis integrable
. . . but in this case the outcome is in fact better.
![Page 66: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/66.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Bourgain property & Birkhoff integrability
1 Given H ⊂ X τp(B) compact and µ Radon probability wehave studied (Pettis) integrability of id : H → X usingBourgain property of
Zid = x∗ id : x∗ ∈ BX ∗ ⊂ RH .
2 In general if (Ω,Σ,µ) is a complete probability space andf : Ω → X is bounded and such that
Zf = x∗ f : x∗ ∈ BX ∗ ⊂ RΩ
has Bourgain property what can you say about f ?
Using techniques of Pettis integration the known answer is: f isPettis integrable. . . but in this case the outcome is in fact better.
![Page 67: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/67.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Bourgain property & Birkhoff integrability
1 Given H ⊂ X τp(B) compact and µ Radon probability wehave studied (Pettis) integrability of id : H → X usingBourgain property of
Zid = x∗ id : x∗ ∈ BX ∗ ⊂ RH .
2 In general if (Ω,Σ,µ) is a complete probability space andf : Ω → X is bounded and such that
Zf = x∗ f : x∗ ∈ BX ∗ ⊂ RΩ
has Bourgain property what can you say about f ?
Using techniques of Pettis integration the known answer is: f isPettis integrable. . . but in this case the outcome is in fact better.
![Page 68: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/68.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Birkhoff definition
Let f : Ω −→ X be a function. If Γ is a partition of Ω intocountably many sets (An) of Σ, the function f is called summablewith respect to Γ if the restriction f |An is bounded wheneverµ(An) > 0 and the set of sums
J(f ,Γ) =
∑n
f (tn)µ(An) : tn ∈ An
is made up of unconditionally convergent series.
The function f is said to be Birkhoff integrable if for every ε > 0there is a countable partition Γ = (An) of Ω in Σ for which f issummable and
‖ ‖−diam(J(f ,Γ)) < ε.
In this case, the Birkhoff integral (B)∫Ω f dµ of f is the only
point in the intersection⋂co(J(f ,Γ)) : f is summable with respect to Γ.
![Page 69: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/69.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Birkhoff definition
Let f : Ω −→ X be a function. If Γ is a partition of Ω intocountably many sets (An) of Σ, the function f is called summablewith respect to Γ if the restriction f |An is bounded wheneverµ(An) > 0 and the set of sums
J(f ,Γ) =
∑n
f (tn)µ(An) : tn ∈ An
is made up of unconditionally convergent series.
The function f is said to be Birkhoff integrable if for every ε > 0there is a countable partition Γ = (An) of Ω in Σ for which f issummable and
‖ ‖−diam(J(f ,Γ)) < ε.
In this case, the Birkhoff integral (B)∫Ω f dµ of f is the only
point in the intersection⋂co(J(f ,Γ)) : f is summable with respect to Γ.
![Page 70: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/70.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Birkhoff definition
Let f : Ω −→ X be a function. If Γ is a partition of Ω intocountably many sets (An) of Σ, the function f is called summablewith respect to Γ if the restriction f |An is bounded wheneverµ(An) > 0 and the set of sums
J(f ,Γ) =
∑n
f (tn)µ(An) : tn ∈ An
is made up of unconditionally convergent series.
The function f is said to be Birkhoff integrable if for every ε > 0there is a countable partition Γ = (An) of Ω in Σ for which f issummable and
‖ ‖−diam(J(f ,Γ)) < ε.
In this case, the Birkhoff integral (B)∫Ω f dµ of f is the only
point in the intersection⋂co(J(f ,Γ)) : f is summable with respect to Γ.
![Page 71: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/71.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Birkhoff integrability: properties
Birkhoff integrability:
1 was introduced in [Bir35].
2 lies strictly between Bochner and Pettis integrability,[Pet38, Phi40]
3 doesn’t involve a barycentric definition;
4 if X is separable Birkhoff=Pettis;
5 Birkhoff integrability has been historically ignored.
Our basic result
We characterize Birkhoff integrability via the property of Bourgain.
![Page 72: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/72.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Birkhoff integrability: properties
Birkhoff integrability:
1 was introduced in [Bir35].
2 lies strictly between Bochner and Pettis integrability,[Pet38, Phi40]
3 doesn’t involve a barycentric definition;
4 if X is separable Birkhoff=Pettis;
5 Birkhoff integrability has been historically ignored.
Our basic result
We characterize Birkhoff integrability via the property of Bourgain.
![Page 73: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/73.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Birkhoff integrability: properties
Birkhoff integrability:
1 was introduced in [Bir35].
2 lies strictly between Bochner and Pettis integrability,[Pet38, Phi40]
3 doesn’t involve a barycentric definition;
4 if X is separable Birkhoff=Pettis;
5 Birkhoff integrability has been historically ignored.
Our basic result
We characterize Birkhoff integrability via the property of Bourgain.
![Page 74: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/74.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Birkhoff integrability: properties
Birkhoff integrability:
1 was introduced in [Bir35].
2 lies strictly between Bochner and Pettis integrability,[Pet38, Phi40]
3 doesn’t involve a barycentric definition;
4 if X is separable Birkhoff=Pettis;
5 Birkhoff integrability has been historically ignored.
Our basic result
We characterize Birkhoff integrability via the property of Bourgain.
![Page 75: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/75.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Birkhoff integrability: properties
Birkhoff integrability:
1 was introduced in [Bir35].
2 lies strictly between Bochner and Pettis integrability,[Pet38, Phi40]
3 doesn’t involve a barycentric definition;
4 if X is separable Birkhoff=Pettis;
5 Birkhoff integrability has been historically ignored.
Our basic result
We characterize Birkhoff integrability via the property of Bourgain.
![Page 76: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/76.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Birkhoff integrability: properties
Birkhoff integrability:
1 was introduced in [Bir35].
2 lies strictly between Bochner and Pettis integrability,[Pet38, Phi40]
3 doesn’t involve a barycentric definition;
4 if X is separable Birkhoff=Pettis;
5 Birkhoff integrability has been historically ignored.
Our basic result
We characterize Birkhoff integrability via the property of Bourgain.
![Page 77: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/77.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Bourgain property and Birkhoff integrability
Theorem (Rodriguez-B.C., 2005)
Let f : Ω → X be a bounded function. TFAE:
(i) f is Birkhoff integrable;
(ii) Zf = 〈x∗, f 〉 : x∗ ∈ BX ∗ has Bourgain property.
Theorem (Rodriguez-B.C., 2005)
Let f : Ω −→ X be a function. TFAE:
(i) f is Birkhoff integrable;
(ii) Zf is uniformly integrable, Zf has Bourgain property.
![Page 78: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/78.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Applications to URL integrable functions
Theorem (Rodriguez-B.C., 2005)
Let f : Ω −→ X be a function. TFAE:
(i) f is Birkhoff integrable;
(ii) there is x ∈ X satisfying: for every ε > 0 there is a countablepartition Γ of Ω in Σ for which f is summable and
‖S(f ,Γ,T )−x‖< ε for every choice T in Γ;
(iii) there is y ∈ X satisfying: for every ε > 0 there is a countablepartition Γ of Ω in Σ such that f is summable with respect toeach countable partition Γ′ finer than Γ and
‖S(f ,Γ′,T ′)−y‖< ε for every choice T ′ in Γ′.
In this case, x = y =∫Ω f dµ.
![Page 79: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/79.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Musia l question?
Zbl 0974.28007Kadets, V.M.; Tseytlin, L.M.On “integration” of non-integrable vector-valued functions.Mat. Fiz. Anal. Geom. 7, No.1, 49-65 (2000)
Let µ be the Lebesgue measure on [0,1] and X be a Banach space. A function f : [0,1]→ X is called absolutelyRiemann-Lebesgue integrable over a measurable set A⊂ [0,1] if there is x ∈ X such that for every ε > 0 there existsa measurable partition 〈∆i 〉∞i=1 of A such that for every finer measurable partition 〈Γi 〉∞j=1 of A and arbitrary points
sj ∈ Γj one has ‖∑j f (sj )µ(Γj )−x‖< ε and ∑j f (sj )µ(Γj ) is absolutely convergent (〈Γi 〉∞j=1 is finer than 〈∆i 〉∞i=1 if
each ∆i is a union of some Γj ’s). In case of unconditional convergence one gets a definition of unconditionallyRiemann-Lebesgue integrable function. . .
There are no results placing ARL and URL integrals among other known typesof integrals such as Birkhoff’s integral or generalized McShane’s integral whichhave similar definitions (and it is relatively easy to see that URL integrablefunctions are also Birkhoff integrable).The rest of the paper is devoted to the study. . .
Kazimierz Musia l (Wroc law)
![Page 80: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/80.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Applications to dual spaces with WRNP
Definition
1 X ∗ has the weak Radon-Nikodym property;
2 X does not contain a copy of `1;
3 for every complete probability space (Ω,Σ,µ) and for everyµ-continuous countably additive vector measure ν : Σ −→ X ∗
of σ -finite variation there is a Pettis integrable functionf : Ω −→ X ∗ such that
ν(E ) =∫E
f dµ
for every E ∈ Σ.
![Page 81: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/81.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Applications to dual spaces with WRNP
Theorem: Musia l,Ryll-Nardzewski, Janicka and Bourgain
Let X be a Banach space. TFAE:
1 X ∗ has the weak Radon-Nikodym property;
2 X does not contain a copy of `1;
3 for every complete probability space (Ω,Σ,µ) and for everyµ-continuous countably additive vector measure ν : Σ −→ X ∗
of σ -finite variation there is a Pettis integrable functionf : Ω −→ X ∗ such that
ν(E ) =∫E
f dµ
for every E ∈ Σ.
![Page 82: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/82.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Applications to dual spaces with WRNP
Rodriguez-B.C. 2005
Let X be a Banach space. TFAE:
1 X ∗ has the weak Radon-Nikodym property;
2 X does not contain a copy of `1;
3 for every complete probability space (Ω,Σ,µ) and for everyµ-continuous countably additive vector measure ν : Σ −→ X ∗
of σ -finite variation there is a Birkhoff integrable functionf : Ω −→ X ∗ such that
ν(E ) =∫E
f dµ
for every E ∈ Σ.
![Page 83: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/83.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The integral for a multifunction
F : Ω −→ cwk(X )
g
G
t0 1
?
6
There are several standard ways of dealing withintegration for F :
1 take a reasonable embedding j from cwk(X )into a Banach space Y (= `∞(BX ∗)) and dealwith the integrability of j F ;
2 take all integrable selectors f of F andconsider∫
F dµ =
∫f dµ : f integra. sel.F
.
1 Debreu, [Deb67], used the embedding technique dealing withBochner integrability;
2 Aumann, [Aum65], used the selector technique;
3 We used the embedding technique with Birkhoff integrability:Rodriguez-B.C., 2004.
![Page 84: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/84.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The integral for a multifunction
F : Ω −→ cwk(X )
g
G
t0 1
?
6
There are several standard ways of dealing withintegration for F :
1 take a reasonable embedding j from cwk(X )into a Banach space Y (= `∞(BX ∗)) and dealwith the integrability of j F ;
2 take all integrable selectors f of F andconsider∫
F dµ =
∫f dµ : f integra. sel.F
.
1 Debreu, [Deb67], used the embedding technique dealing withBochner integrability;
2 Aumann, [Aum65], used the selector technique;
3 We used the embedding technique with Birkhoff integrability:Rodriguez-B.C., 2004.
![Page 85: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/85.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The integral for a multifunction
F : Ω −→ cwk(X )
g
G
t0 1
?
6f
There are several standard ways of dealing withintegration for F :
1 take a reasonable embedding j from cwk(X )into a Banach space Y (= `∞(BX ∗)) and dealwith the integrability of j F ;
2 take all integrable selectors f of F andconsider∫
F dµ =
∫f dµ : f integra. sel.F
.
1 Debreu, [Deb67], used the embedding technique dealing withBochner integrability;
2 Aumann, [Aum65], used the selector technique;
3 We used the embedding technique with Birkhoff integrability:Rodriguez-B.C., 2004.
![Page 86: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/86.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The integral for a multifunction
F : Ω −→ cwk(X )
g
G
t0 1
?
6f
There are several standard ways of dealing withintegration for F :
1 take a reasonable embedding j from cwk(X )into a Banach space Y (= `∞(BX ∗)) and dealwith the integrability of j F ;
2 take all integrable selectors f of F andconsider∫
F dµ =
∫f dµ : f integra. sel.F
.
1 Debreu, [Deb67], used the embedding technique dealing withBochner integrability;
2 Aumann, [Aum65], used the selector technique;
3 We used the embedding technique with Birkhoff integrability:Rodriguez-B.C., 2004.
![Page 87: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/87.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The integral for a multifunction
F : Ω −→ cwk(X )
g
G
t0 1
?
6f
There are several standard ways of dealing withintegration for F :
1 take a reasonable embedding j from cwk(X )into a Banach space Y (= `∞(BX ∗)) and dealwith the integrability of j F ;
2 take all integrable selectors f of F andconsider∫
F dµ =
∫f dµ : f integra. sel.F
.
1 Debreu, [Deb67], used the embedding technique dealing withBochner integrability;
2 Aumann, [Aum65], used the selector technique;
3 We used the embedding technique with Birkhoff integrability:Rodriguez-B.C., 2004.
![Page 88: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/88.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
The integral for a multifunction
F : Ω −→ cwk(X )
g
G
t0 1
?
6f
There are several standard ways of dealing withintegration for F :
1 take a reasonable embedding j from cwk(X )into a Banach space Y (= `∞(BX ∗)) and dealwith the integrability of j F ;
2 take all integrable selectors f of F andconsider∫
F dµ =
∫f dµ : f integra. sel.F
.
1 Debreu, [Deb67], used the embedding technique dealing withBochner integrability;
2 Aumann, [Aum65], used the selector technique;
3 We used the embedding technique with Birkhoff integrability:Rodriguez-B.C., 2004.
![Page 89: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/89.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Kadets-Rodriguez-B.C., 2006
X For X separable, we study the Pettis integral formulti-functions F : Ω → cwk(X );
X From the notion of Pettis integrability for such an F studiedin the literature one readily infers that if we embedj : cwk(X ) → `∞(BX ∗) then j F is integrable with respect toa norming subset of B`∞(BX∗ )∗ .
X A natural question arises: When is j F Pettis integrable?
X Pettis integrability of any cwk(X )-valued function F isequivalent to the Pettis integrability of j F if and only if Xhas the Schur property. . .
X . . . if and only if equivalent to the fact that cwk(X ) isseparable when endowed with the Hausdorff distance.
![Page 90: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/90.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Kadets-Rodriguez-B.C., 2006
X For X separable, we study the Pettis integral formulti-functions F : Ω → cwk(X );
X From the notion of Pettis integrability for such an F studiedin the literature one readily infers that if we embedj : cwk(X ) → `∞(BX ∗) then j F is integrable with respect toa norming subset of B`∞(BX∗ )∗ .
X A natural question arises: When is j F Pettis integrable?
X Pettis integrability of any cwk(X )-valued function F isequivalent to the Pettis integrability of j F if and only if Xhas the Schur property. . .
X . . . if and only if equivalent to the fact that cwk(X ) isseparable when endowed with the Hausdorff distance.
![Page 91: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/91.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Kadets-Rodriguez-B.C., 2006
X For X separable, we study the Pettis integral formulti-functions F : Ω → cwk(X );
X From the notion of Pettis integrability for such an F studiedin the literature one readily infers that if we embedj : cwk(X ) → `∞(BX ∗) then j F is integrable with respect toa norming subset of B`∞(BX∗ )∗ .
X A natural question arises: When is j F Pettis integrable?
X Pettis integrability of any cwk(X )-valued function F isequivalent to the Pettis integrability of j F if and only if Xhas the Schur property. . .
X . . . if and only if equivalent to the fact that cwk(X ) isseparable when endowed with the Hausdorff distance.
![Page 92: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/92.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Kadets-Rodriguez-B.C., 2006
X For X separable, we study the Pettis integral formulti-functions F : Ω → cwk(X );
X From the notion of Pettis integrability for such an F studiedin the literature one readily infers that if we embedj : cwk(X ) → `∞(BX ∗) then j F is integrable with respect toa norming subset of B`∞(BX∗ )∗ .
X A natural question arises: When is j F Pettis integrable?
X Pettis integrability of any cwk(X )-valued function F isequivalent to the Pettis integrability of j F if and only if Xhas the Schur property. . .
X . . . if and only if equivalent to the fact that cwk(X ) isseparable when endowed with the Hausdorff distance.
![Page 93: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/93.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Kadets-Rodriguez-B.C., 2006
X For X separable, we study the Pettis integral formulti-functions F : Ω → cwk(X );
X From the notion of Pettis integrability for such an F studiedin the literature one readily infers that if we embedj : cwk(X ) → `∞(BX ∗) then j F is integrable with respect toa norming subset of B`∞(BX∗ )∗ .
X A natural question arises: When is j F Pettis integrable?
X Pettis integrability of any cwk(X )-valued function F isequivalent to the Pettis integrability of j F if and only if Xhas the Schur property. . .
X . . . if and only if equivalent to the fact that cwk(X ) isseparable when endowed with the Hausdorff distance.
![Page 94: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/94.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
. . . back to boundaries: Simons’ techniques
Theorem: Orihuela, Munoz, B.C., to appear
Let J : X → 2BX∗ be the duality mapping
J(x) := x∗ ∈ BX ∗ : x∗(x) = ‖x‖.
TFAE:
(i) X is Asplund, i.e., X ∗ has RNP;
(ii) for some fixed 0 < ε < 1, J has an ε-selector f that sendsnorm separable subsets of X into norm separable subsets ofX ∗;
(iii) for some fixed 0 < ε < 1, dual unit ball BX ∗ is normε-fragmented.
ε-selector: d(f (x),J(x)) < ε for every x ∈ X
![Page 95: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/95.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
Two. . . three nice problems
1 The boundary problem in full generality (Godefroy).
2 Characterize Banach spaces X for which (BX ∗ ,w∗) issequentially compact (Diestel).
3 Characterize Banach spaces X for which (BX ∗ ,w∗) is angelic.
![Page 96: Topology, measure theory and Banach spaces Semevi...B. C and J. Rodr´ıguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), no. 2, 540–560, Special](https://reader033.vdocuments.us/reader033/viewer/2022041817/5e5bd5eb47db77647b2b27b1/html5/thumbnails/96.jpg)
Bourgain property & compactness Bourgain property & Birkhoff integrability Aumman&Debreu&Pettis integrals multifunctions
References
R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12(1965), 1–12. MR 32 #2543
G. Birkhoff, Integration of functions with values in a Banach space, Trans.Amer. Math. Soc. 38 (1935), no. 2, 357–378. MR 1 501 815
G. Debreu, Integration of correspondences, Proc. Fifth Berkeley Sympos.Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II:Contributions to Probability Theory, Part 1, Univ. California Press,Berkeley, Calif., 1967, pp. 351–372. MR 37 #3835
B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44(1938), no. 2, 277–304. MR 1 501 970
R. S. Phillips, Integration in a convex linear topological space, Trans.Amer. Math. Soc. 47 (1940), 114–145. MR 2,103c
L. H. Riddle and E. Saab, On functions that are universally Pettisintegrable, Illinois J. Math. 29 (1985), no. 3, 509–531. MR 86i:28012