boolean algebras: convergence and measure

Topology and its Applications 111 (2001) 139–149

Boolean algebras: Convergence and measure✩

Roman Fric

Matematický Ústav SAV, Grešákova 6, 040 01 Košice, Slovakia

Received 1 July 1999; received in revised form 4 July 1999

Abstract

We study topological and categorical aspects of the extension ofσ -additive measures from a fieldof sets to the generatedσ -field within a category of Boolean algebras carrying initial sequentialconvergences with respect to2-valued homomorphisms. We describe the relationship betweenσ -additivity and sequential continuity of finitely additive measures. A key role is played by theepireflective subcategory of absolutely sequentially closed objects. In case of fields of sets suchobjects are exactlyσ -fields. The results provide information about basic notions of probabilitytheory: events, probability measures, and random functions. 2001 Elsevier Science B.V. All rightsreserved.

Keywords:Boolean algebra; Initial sequential convergence; Measure; Sequential continuity;σ -additivity; Field of sets; Extension of measures; Absolutely sequentially closed objects;Epireflective subcategory; Duality;s-perfectness; Measurable space; Measurable map; Field ofprobability events; Probability; Random variable

AMS classification:Primary 54A20; 54C20; 54B30; 60A10, Secondary 28A60; 06E15; 18B99

Introduction

The idea that the extension of probability measures from a field of events to the generatedσ -field is of a topological nature is due to Novák. The outcome is the theory of sequentialenvelopes outlined at the First Prague Topological Symposium in 1961 (see [11]) andpresented in [12]. It was Hušek who in 1971 (an unpublished manuscript) pointed out thecategorical background of the sequential envelope: it is an epireflection of a sequentialconvergence space belonging to the category simply generated by the closed interval[0,1] or the real lineR to the subcategory of absolutely sequentially closed spaces. Theconstruction has been generalized to quite arbitrary classes of real-valued functions in [13]

✩ Supported by VEGA Grant no. 5125/99.E-mail address:[email protected] (R. Fric).

0166-8641/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0166-8641(99)00195-9

140 R. Fric / Topology and its Applications 111 (2001) 139–149

and applied to the extension of probabilities: for each field of sets, the generatedσ -field isits sequential envelope with respect to the set of all probability measures. Generalizationsof the sequential envelope appeared in [7] and the categorical background can be foundin [14].

Let us explain the notion of the absolute sequential closedness. LetY be a set carryingthe initial sequential convergenceL with respect to a classC0 of sequentially continuousreal-valued functions onY , i.e., a sequence〈yn〉 converges toy underL iff f (yn)→ f (y)

for all f ∈ C0. In extendingC0 we are looking for a setY ′ carrying a sequential convergenceL′ such that:

(i) Y 6⊆ Y ′, L′ restricted toY is equal toL, andY ′ is the smallest sequentially closedsubset ofY ′ containingY ;

(ii) eachf ∈ C0 can be uniquely extended to a sequentially continuous function onY ′andL′ is the initial sequential convergence with respect to the extensions.

If there is no suchY ′ andL′, thenY carryingL is said to beC0-sequentially complete.Observe that it amounts to the following: a sequence〈yn〉 in Y converges underL iff foreachf ∈ C0 the numerical sequence〈f (yn)〉 converges (cf. [4]). We shall call such spacesabsolutely sequentially closedwith respect to the extension ofC0. If no confusion can arise,we shall speak of absolutely sequentially closed spaces.

Recall that a sequential convergence on a setY can be considered as a subsetL ofYN × Y satisfying the usual four axioms of convergence:

(i) each constant sequence〈y〉 converges toy;(ii) if 〈yn〉 converges toy, then each subsequence〈y ′n〉 of 〈yn〉 converges toy;(iii) the uniqueness of limits;(iv) the Urysohn axiom.

If (〈yn〉, y) ∈ L, then we say that the sequence〈yn〉 converges toy (underL).If Y carries algebraic operations, thenL is said to be compatible if each operation is

sequentially continuous.Let A be a field of subsets ofX. Then A carries a natural Boolean structure:⋃,⋂,∅,X, c, and anatural sequential convergence: 〈An〉 converges toA iff

A=∞⋃k=1

∞⋂n=k

An =∞⋂k=1

∞⋃n=k

An

and compatibility means: if〈An〉 converges toA and〈Bn〉 converges toB, then〈An ∪Bn〉converges toA∪B, 〈An ∩Bn〉 converges toA∩B, 〈X \An〉 converges toX \A. Clearly,eachx ∈X represents a sequentially continuous homomorphism ofA into the two-elementBoolean algebra2. Observe that the convergence onA is initial with respect to all suchhomomorphisms. In the field2X of all subsets ofX, the generatedσ -algebraσ(A) is thesmallest sequentially closed subset containingA. If not indicated otherwise, each field ofsets carries the natural sequential convergence.

More information about sequential convergences and compatible sequential conver-gences can be found in [12,5]. Standard references on Boolean algebras and category the-ory are [15,6], respectively.

R. Fric / Topology and its Applications 111 (2001) 139–149 141

Let A be a Boolean algebra. By aStone familyof A we understand a subsetH ofthe set of all homomorphisms ofA into 2 such that ifA,B ∈ A andA 6= B, then thereexistsh ∈H such thath(A) 6= h(B). Each Stone familyH of A induces onA the initialconvergenceLH : a sequence〈An〉 converges toA underLH iff 〈h(An)〉 converges in2to h(A) for eachh ∈ H . Such convergences are called2-generated. If A is a Booleanalgebra andL is a2-generated convergence, thenA carryingL is said to be2-generated.Let B(2) be the category whose objects are2-generated Boolean algebras and whosemorphisms are sequentially continuous Boolean homomorphisms. If(A,L) is 2-generated,then hom(A,L) denotes the set of all morphisms of(A,L) into 2. The subcategoryAB(2) consisting of absolutely sequentially closed objects with respect to the extensionof hom(A,L) is epireflective inB(2) (cf. [3]). Let (A,L) ∈ B(2). Observe that “allinformation” about(A,L) is contained in its embedding into the field2hom(A,L) of allsubsets ofhom(A,L) via the natural evaluation map.

1. Sequentially continuous measures

This section is devoted to sequentially continuous measures on objects ofB(2). Weshow that the epireflection ofB(2) intoAB(2) plays a key role in the relationship betweensequential continuity andσ -additivity of measures.

Let A be a Boolean algebra. By ameasureonA we understand a finite, nonnegativeand finitely additive function. We say that a measurem is σ -additiveif whenever〈An〉 is asequence of disjoint elements ofA and

∨∞n=1An exists, then

m

( ∞∨n=1

An

)=∞∑n=1

m(An).

Let L be a2-generated convergence onA. If m(An)→ 0 for each monotone sequence〈An〉 converging to0 underL, then we say thatm is L-continuous from above at0.

In [10] it is proved that, for rings of sets, eachσ -additive (observe that, in a ring ofsets,

∨∞n=1An can differ from

⋃∞n=1An and hence we should speak ofσ∨-additivity and

σ∪-additivity; for σ -rings the two notions coincide and, otherwise, it will follow from thecontext whether∨ or ∪ is meant) bounded measure is sequentially continuous. The resultcan be extended toB(2).

Proposition 1.1. Let (A,L) ∈ B(2) and letm be a measure onA. If m is L-continuousfrom above at0, thenm is sequentially continuous.

Proof. (1) First, let 〈An〉 be an sequence inA converging to0. We shall prove thatm(An)→ 0. Let f be a map ofN into N such thatf (k) > k for all k ∈ N. DefineSk =∨f (k)

i=k Ai , k ∈N. Sincem is monotone and

Sn 6(

n∧k=1

Sk

)∨(n−1∨k=1

(Sk+1 \ Sk)),


we get

m(Sn)6m(

n∧k=1

Sk

)+n−1∑k=1

m(Sk+1 \ Sk).

Observe that〈∧nk=1Sk〉 is a monotone sequence converging to0 and hence the monotone

sequence〈m(∧mk=1Sk)〉 converges to 0. ThenAn 6 Sn implies

limsupm(An)6∞∑k=1

m(Sk+1 \ Sk). (∗)

Now, let ε be a positive real number. We claim that a suitable choice off makes theright-hand side of (∗) smaller thanε. Indeed, for eachk ∈ N, the sequence〈∨k−1+n

i=k Ai〉is monotone and the bounded monotone sequence〈m(∨k−1+n

i=k Ai)〉 converges. Then, foreachk ∈N, there existsf (k)> k such that for each natural numberp, p> f (k), we have

m

((p∨i=kAi

)\(f (k)∨i=k

Ai

))=m

(p∨i=kAi

)−m

(f (k)∨i=k

Ai

)< ε2−k.

Since

Sk+1 \ Sk 6((

p∨i=kAi

)\ Sk

)wheneverp> f (k + 1), it follows that

∞∑i=1

m(Sk+1 \ Sk) < ε∞∑i=1

2−k = ε. (∗∗)

Thusm(An)→ 0.(2) Now, let 〈En〉 be a sequence converging inA to E. We have to prove that

m(En)→m(E). ClearlyE = (E ∧En) ∨ (E \En) andEn = (En ∧ E)∨ (En \E), andthe sequences〈E \ En〉 and〈En \ E〉 converge inA to 0. Accordingly,m(E \ En)→ 0andm(En \E)→ 0. Fromm(E)=m(E ∧En)+m(E \En) we getm(E ∧En)→m(E)

and fromm(En)=m(En ∧E)+m(En \E) we getm(En)→m(E). This completes theproof. 2Example 1.2. LetA be the Boolean algebra of subsets of(0,1) generated by all intervalsof the form[a, b)⊂ (0,1). Each pointx ∈ (0,1) generates a homomorphismhx of A into2 by puttingh(A)= 1 if x ∈ A andh(A) = 0 otherwise. Further, for eachA ∈ A, defineh0(A)= 1 if A contains an interval of the form(0, a), a ∈ (0,1), andh0(A)= 0 otherwise.Thenh0 is a homomorphism ofA into 2 andH = {hx; x ∈ [0,1)} is a Stone family ofA.Let LH be the corresponding2-generated convergence onA. Observe thath0 is anLH -continuous measure onA which fails to beσ -additive. Indeed,〈[1/(n+ 1),1/n)〉 is a dis-joint sequence,(0,1)=∨∞n=1[1/(n+ 1),1/n) in A, but

∑∞n=1h0([1/(n+ 1),1/n))= 0.

Lemma 1.3. Let (A,L) ∈ B(2) and let〈An〉 be a sequence inA.(i) For each h ∈ hom(A,L), the sequences〈∨n

i=1h(Ai)〉 and 〈∧ni=1h(Ai)〉 are

convergent sequences in2.


(ii) If the sequence〈∨ni=1Ai〉 is L-converging toA, respectively the sequence

〈∧ni=1Ai〉 is L-converging toB, thenA=∨∞i=1Ai , respectivelyB =∧∞i=1Ai .

(iii) If (A,L) ∈ AB(2), thenA is σ -complete and the sequence〈∨ni=1Ai〉 is L-

converging to∨∞i=1Ai , respectively the sequence〈∧n

i=1Ai〉 is L-converging to∧∞i=1Ai .

Proof. (i) Follows from the fact that both sequences are monotone.(ii) Observe that forX,Y ∈A we haveX 6 Y iff h(X) 6 h(Y ) for all h ∈ hom(A,L).

Since〈∨ni=1Ai〉 isL-converging toA, necessarilyh(Ai)6 h(A) for eachh ∈ hom(A,L).

HenceAi 6 A for all i ∈ N. Now, let A′ ∈ A and let Ai 6 A′ for all i ∈ N. Leth ∈ hom(A,L). If h(A′)= 0, thenh(

∨ni=1Ai)= 0 for all n ∈N; hence 0= h(A)6 h(A′).

On the other handh(A)6 h(A′), if h(A′)= 1. ThusA6A′ andA=∨∞i=1Ai . The proofof B =∧∞i=1Ai is analogous and it is omitted.

(iii) Consider(A,L) embedded into2hom(A,L) via the natural evaluation. Since(A,L)is absolutely sequentially closed,A is sequentially closed. Now, the assertion follows from(i) and (ii). This completes the proof.2Proposition 1.4. LetA be a Boolean algebra and letm be a measure onA. If m is σ -additive, thenm is sequentially continuous in each2-generated convergence onA.

Proof. Let L be a2-generated convergence onA. It follows from Proposition 1.1 that itsuffices to prove theL-continuity ofm from above at0.

Let 〈Bn〉 be a monotone sequenceL-converging to0. PutAn = Bn \ Bn+1, n ∈ N.Since〈B1 \ Bn〉 converges toB1 and

∨n−1i=1 Ai = B1 \ Bn, n = 2,3, . . . , it follows from

Lemma 1.3 that∨∞i=1Ai = B1. Fromm(

∨n−1i=1 Ai) =

∑n−1i=1 m(Ai) = m(B1) − m(Bn)

and from theσ -additivity of m it follows that the sequence〈m(Bn)〉 converges to 0. Thiscompletes the proof.2

As observed by the referee, the converse implication holds, too.

Proposition 1.5. LetA be a Boolean algebra and letm be a measure onA. If m is L-continuous from above at0 in each2-generated convergenceL onA, thenm is σ -additive.

Proof. Clearly,m is σ -additive iff m(An)→ 0 whenever〈An〉 is a decreasing sequencesuch that0=∧∞n=1An. Now, let 〈An〉 be a decreasing sequence such that0=∧∞n=1An.LetH be the set of all homomorphisms ofA into 2 such thath(An)= 1 for at most finitelymanyn ∈N. ThenH is a Stone family ofA and〈An〉 converges to0 under the2-generatedsequential convergenceLH . Assuming thatm is LG-continuous for each Stone familyGof A, it follows thatm(An)→ 0. Consequently,m is σ -additive. 2Proposition 1.6. Let (A,L) ∈AB(2) and letm be a measure onA. If m isL-continuous,thenm is σ -additive.

Proof. The assertion follows from Lemma 1.3.2


As pointed out by the referee, Proposition 1.6 can be converted to get a characterizationof the categoryAB(2). He suggested some answers to the following

Question 1. What are the objects of the categoryAB(2)?

Question 2. Under what conditions for a given Boolean algebraA there exists a2-generated convergenceL such that(A,L) ∈AB(2)?

Some of the suggestions follow from an analysis of the embedding of(A,L) ∈ B(2) intoits epireflection(A,L) ∈ AB(2) (cf. [3]) and from Lemma 1.3. We canonically embed(A,L) into the power2hom(A,L) carrying the pointwise convergence and, identifyingfor simplicity (A,L) with its image, we get(A,L) as the smallest sequentially closedsubobject of2hom(A,L) containingA. Hence(A,L) ∈AB(2) iff A is sequentially closed.Observe that, in fact,2hom(A,L) is the field of all subsets ofhom(A,L) and (A,L) is asubfield and(A,L) is the generatedσ -field.

Corollary 1.7. Let (A,L) ∈ B(2). Then the following are equivalent:(i) (A,L) ∈AB(2);(ii) Each increasing sequence inA is convergent underL;(iii) A is σ -complete and each increasing sequence〈An〉 converges underL to∨∞

n=1An;(iv) A is σ -complete and each decreasing sequence〈An〉 such that0 = ∧∞n=1An

converges underL to 0;(v) There exists aσ -algebraA of sets and a Boolean isomorphismh ofA ontoA such

that bothh and its inverse are sequentially continuous.

Proposition 1.8. Let (A,L) ∈ B(2). Then the following are equivalent:(i) (A,L) ∈AB(2);(ii) A is σ -complete and each measureL-continuous from above at0 is σ -additive;(iii) A is σ -complete and eachh ∈ hom(A,L) is aσ -additive measure;(iv) A is σ -complete andL is the initial convergence with respect to theσ -additive{0,1}-valued measures onA.

Proof. According to Proposition 1.6, (i) implies (ii). Trivially, (ii) implies (iii) and (iii)implies (iv). It remains to prove that (iv) implies (i).

Assume (iv), i.e.,A is σ -complete and a sequence〈An〉 converges toA underL iff foreachσ -additive{0,1}-valued measurem onA the sequence〈m(An)〉 converges tom(A).Let 〈An〉 be an increasing sequence inA. Since for eachσ -additive{0,1}-valued measurem onA the sequence〈m(An)〉 converges tom(

∨∞n=1An), the sequence〈An〉 converges

underL to∨∞n=1An. Hence (i) follows by Corollary 1.7. 2

Corollary 1.9. LetA be a Boolean algebra. Then the following are equivalent:(i) There exists a2-generated convergenceL onA such that(A,L) ∈AB(2);(ii) A is σ -complete andσ -additive{0,1}-valued measures form a Stone family ofA.


2. Fields of sets revisited

Let FS be the category whose objects are fields of sets carrying the natural sequentialconvergence and whose morphisms are sequentially continuous Boolean homomorphisms.If A is a field of sets, thenhom(A) will denote the set of all morphisms ofA into 2.

Let A be a field of subsets ofX. Recall that the generatedσ -field σ(A) is the smallestsequentially closed subfield of the field2X of all subsets ofX which containsA. Further,if two sequentially continuous maps ofσ(A) into a space having unique sequential limitsagree onA, then they are identical, cf. Lemma 5 in [12]. In particular, each sequentiallycontinuous measure onA has a unique extension overσ(A).

Lemma 2.1. Let A be a field of subsets ofX and leth ∈ hom(A). Then there exists auniqueh ∈ hom(σ (A)) such thath �A=h.

Proof. Clearly, h is a σ -additive measure onA and hence it can be uniquely extendedto aσ -additive measureh on the generatedσ -field σ(A). It follows from Proposition 1.4thath is sequentially continuous. SinceA is sequentially dense inσ(A), h is a two-valuedmeasure and henceh ∈ hom(σ (A)). 2Proposition 2.2. Let A be a field of subsets ofX. ThenA = σ(A) iff A is absolutelysequentially closed with respect to the extension of hom(A).

Proof. (1) Let A = σ(A). If 〈An〉 is a sequence inA converging in some admissibleextensionA′ of A, then for eachh ∈ hom(A) the sequence〈h(An)〉 converges in2. Thusthe sequence〈An〉 converges in2X. SinceA = σ(A) is sequentially closed in2X, thesequence〈An〉 converges inA. HenceA is sequentially closed also inA′ and, consequently,absolutely sequentially closed.

(2) Let A be absolutely sequentially closed. It follows from Lemma 2.1 thatA issequentially closed inσ(A) and henceA= σ(A). 2

Let AFSbe the subcategory ofFSconsisting ofσ -fields of sets.

Proposition 2.3. LetA be a field of subsets ofX and leth be a sequentially continuoushomomorphism ofA into a σ -fieldB of subsets ofY . Thenh can be uniquely extended toa sequentially continuous homomorphismh of σ(A) into B.

Proof. To simplify the notation, we identifyB with its image under the natural evaluationmap into2hom(B). ThenB is a sequentially closed subspace of2hom(B), h can be consideredas a morphism into the product

∏f∈hom(B) 2f , and each projectionπf yields a morphism

πf ◦h of A into 2f . According to Proposition 2.1 eachπf ◦h can be uniquely extended toa morphismπf ◦ h of σ(A) into 2f . But

∏f∈hom(B) 2f is the categorical product and

hence there is a uniquely determined morphismh of σ(A) into the product such thatπf ◦ h= πf ◦ h for eachf ∈ hom(B). SinceA is sequentially dense inσ(A), the restrictionof h toA is equal toh. This completes the proof.2


It follows from Proposition 2.3 that the embeddingA ↪→ σ(A) yields a functorσ fromFS into AFS.

Corollary 2.4. σ : FS→ AFS is an epireflector.

LetA be a reduced field of subsets ofX. In what follows, all fields will be reduced. PutX∗ = hom(A). ForA ∈A, defineA∗ = {h ∈X∗; h(A)= 1}. ThenA∗ = {A ∈A} is a fieldof subsets ofX∗. SinceX can be considered as a subset ofX∗, A can be considered as thetrace ofA∗ ontoX. This yields an isomorphism betweenA andA∗ as objects ofFS. Ingeneral,X $ X∗ and each morphism inhom(A∗) can be represented by a point ofX∗. IfX =X∗, thenA is said to bes-perfect. There is a one-to-one correspondence betweenX∗and ultrafilters onA having the countable intersection property (CIP), cf. [1,2].

By a measurable spacewe understand a pair(X,A), whereA is a field of subsets ofX.Sets inA are calledmeasurable.

Let MM be the category whose objects are measurable spaces and whose morphisms aremeasurable maps. LetSPMMbe the subcategory consisting of measurable spaces(X,A)such thatA is s-perfect. As shown in [2], categoriesB(2) and the dual categorySPMMop

are naturally equivalent. The duality is based on the fact that each measurable map inducesa sequentially continuous Boolean homomorphism (going the opposite direction) and eachsequentially continuous Boolean homomorphism of ans-perfect field of sets is induced bya unique measurable map (going the opposite direction). This duality generalizes the usualStone duality between Boolean algebras (carrying the initial sequential convergence withrespect to all homomorphisms into2) and perfect fields of sets.

Proposition 2.5. Let f be a measurable map of a measurable space(X,A) into ameasurable space(Y,B). If B is s-perfect, then there exists a unique measurable mapf ∗ of (X∗,A∗) into (Y,B) such thatf ∗ �X = f .

Proof. Sincef← is a sequentially continuous Boolean homomorphism ofB into A andthe embeddinge of X into X∗ induces a sequentially continuous Boolean isomorphisme← of A ontoA∗, it follows from thes-perfectness ofB that the compositione← ◦ f← isinduced by a unique measurable mapf ∗ of (X∗,A∗) into (Y,B) such thatf ∗ �X= f . 2

For each object(X,A) of MM, the embedding ofX intoX∗ (observe thatX is dense in(X∗,A∗) in the sense that for each nonemptyA ∈A∗ there existsx ∈X∗ such thatx ∈A)yields a functors : MM→ SPMM.

Corollary 2.6. s : MM→ SPMM is an epireflector.

Let AMM be the subcategory ofMM consisting of objects(X,A) such thatA= σ(A).

Proposition 2.7. AMM is a monocoreflective subcategory of MM.


Proof. Let (X,A) be a measurable space. The identity map of(X,σ(A)) onto (X,A) isa monomorphism. Iff is a measurable map of a measurable space(Y,σ (B)) into (X,A),thenf is a measurable map of(Y,σ (B)) into (X,σ(A)) (the uniqueness is trivial).2

We close this section with few more properties ofs-perfect fields. Proposition 2.7 in [2]states that if a fieldA of subsets iss-perfect, then the generatedσ -field σ(A) is s-perfect,too.

Corollary 2.8. The composition ofs andσ is commutative.

Proposition 2.9. Let T 6= ∅ and, for eacht ∈ T , letAt be ans-perfect field of subsets ofXt . Then the product field

∏t∈T At on

∏t∈T Xt is s-perfect.

Proof. LetF be an ultrafilter in∏t∈T At having the CIP. We have to prove thatF is fixed.

For eachs ∈ T , denote

Fs ={F ∈As;

(F ×

∏t∈(T \{s})

Xt

)∈F

}.

Then eachFt is a filter inAt and sets∏t∈T Ft , whereFt ∈ Ft andFt = Xt for all but

finitely manyt ∈ T , form a base ofF . Moreover, eachFt is an ultrafilter inAt and has theCIP, hence it is fixed. ThusF is fixed. 2Corollary 2.10. Let T 6= ∅ and, for eacht ∈ T , let At be ans-perfectσ -field of sets onXt . Then the productσ -fieldσ(

∏t∈T At ) on

∏t∈T Xt is s-perfect.

Proposition 2.11. LetT 6= ∅ and letBT be theσ -field of all Borel sets inRT . ThenBT iss-perfect.

Proof. It is easy to verify that the minimal field of all subsets ofR which contains allintervals[a,∞), a ∈ R, is s-perfect. The assertion follows from previous propositions.2Corollary 2.12. Let h be a sequentially continuous Boolean homomorphism of the Borelσ -fieldBT of subsets ofRT into a σ -fieldA of subsets ofX. Then there exists a uniquemeasurable mapf of (X,A) into (RT ,BT ) inducingh.

3. Concluding remarks

The results of the previous sections provide information about the basic notions ofprobability theory: events, probability measures, and random variables.

The Boolean approach to fields of events has been thoroughly discussed by Łos in [8]and in [9] he developed a construction in which the elementary events (i.e., ultrafilters)are a prime notion and the field of events is built and some properties of probabilities (e.g.,σ -additivity) are decided only after a particular choice of elementary events is made (based


on data “not supplied by the calculus of probability” and “it should result somehow fromthe physical conditions of the phenomena under examination”).

Construction. Let Z be a nonempty set and letA be a separating family of subsets ofZ, i.e., if A,B ∈ Z andA 6= B, then there existsa ∈A such that eitherA ∈ a andB /∈ a,or B ∈ a andA /∈ a. This yields an evaluation mapϕ of Z into {0,1}A sending eachA ∈ Z to ϕ(A) = {a ∈ A; A ∈ a}. Now, letE(Z,A) be the smallest Boolean algebra ofsubsets ofA containing the imageϕ(Z) of Z. The interpretation is thatZ is an initialset of “important” events (not necessarily a field) and the elements ofA represent theirelementary attributes,E(Z,A) represents the generated field of events as subsets ofA.Clearly,A can be injectively mapped into the setU of all ultrafilters onE(Z,A), i.e.,Boolean homomorphisms into2. A different separating familyA′ of subsets ofZ canyield another Boolean algebraE(Z,A′) of subsets ofA′, but the two algebras can beisomorphic. In fact, the choice ofA not only determines the algebraic structure ofE(Z,A)but it yields “a condition which essentially restricts permissible (probability) functions”,cf. p. 132 in [9].

Remark 3.1. The elementary attributesA induce onA = E(Z,A) a 2-generatedconvergenceL and hence we get an object(A,L) of B(2). In probability we needσ -complete fields of events and the epireflection(A,L) of (A,L) into the subcategoryAB(2)is exactly what is needed:

(i) ForX =A, (A,L) can be represented as a fieldA of subsets ofX, or as a fieldA∗of subsets ofX∗, and(A,L) can be represented as the generatedσ -field σ(A) onX, or σ(A∗) onX∗.

(ii) Each sequentially continuous measure on(A,L) can be uniquely extended to asequentially continuous measure on(A,L) and (A,L) is absolutely sequentiallyclosed with respect to the extensions. Also, the elementary attributesX =A onAcan be uniquely extended to elementary attributes onσ(A) andσ(A) is absolutelysequentially closed with respect to the extensions.

Remark 3.2. The probabilities on(A,L) can be defined as the sequentially continuousnormed measures. Then(A,L) and (A,L) have the same probabilities and bothLandL are initial convergences with respect to the probabilities. For objects ofAB(2),the probabilities areσ -additive. Since the elementary attributes determineL, they alsodetermine which (normed) measure becomes sequentially continuous. Observe that theelementary attributes are{0,1}-valued probabilities.

Remark 3.3. The duality betweenB(2) and SPMM and the properties of functorsσands provide a mathematical bridge between random functions (i.e., families of randomvariables) and morphisms ofB(2).


Acknowledgement

The author expresses his thanks to Gabriele Greco for valuable suggestions leading tothe improvement of the section on sequentially continuous measures.

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boolean algebras: convergence and measure

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