Homework # 2.

1. Let J be a collection of non-degenerate closed intervals on R, andlet A be the set of centers of all intervals in J . Assume that for every > 0and every center x A there exists an interval I J centered at x with|I| < . Finally, assume that

D = sup[|I| : I J ] < +.Show that there exist two sub-collections F1 and F2 of intervals in J sothat each F1,2 is a countable collection of pairwise disjoint intervals and theset A is covered by F1 F2.

2. Let be a Borel measure on R and F any collection of non-degenerateclosed intervals. Let A be the set of centers of the intervals in F . Assumethat (A) < + and that for every > 0 and every center x A thereexists an interval I J centered at x with |I| < . Then for every open setU R there exists a pairwise disjoint countable sub-collection J of intervalsin F so that

IJI U,

and(AU \

( IJ

I))

= 0.

3. Construct a monotone function that is discontinuous on a dense set on[0, 1].

4. Let Ek be a sequence of measurable sets such thatk=1

(Ek) < +.

Show that then almost all x lie in at most finitely many of the sets Ek.5. Let be a non-negative continuous function on Rn such that

= 1.

Given t > 0 define t(x) = tn(x/t). Show that if g C(Rn) with

compact support then

t(g) =

Rnt(x)g(x)dx g(0).

Because of that t is called an approximation of identity. How much canyou weaken the regularity assumptions on and g?

1