6.9. EXERCISES 155

14. There is a monumentally important theorem called the Borel Cantelli lemma. It saysthe following. If you have a measure space (Ω,F ,µ) and if {Ei} ⊆ F is suchthat ∑

∞i=1 µ (Ei) < ∞, then there exists a set N of measure 0 (µ (N) = 0) such that if

ω /∈ N, then ω is in only finitely many of the Ei. Hint: You might look at the set ofall ω which are in infinitely many of the Ei. First explain why this set is of the form∩∞

n=1∪k≥n Ek.

15. Let (Ω,F ,µ) be a measure space. A sequence of functions { fn} is said to convergein measure to a measurable function f if and only if for each ε > 0,

limn→∞

µ ({ω : | fn (ω)− f (ω)|> ε}) = 0

Show that if this happens, then there exists a subsequence{

fnk

}and a set of measure

N such that if ω /∈ N, thenlimk→∞

fnk (ω) = f (ω) .

Also show that if µ is finite and limn→∞ fn (ω)= f (ω) , then fn converges in measureto f .

16. Let N be the positive integers and let F denote the set of all subsets of N. Explainwhy N is a σ algebra. You could let µ (S) be the number of elements of S. This iscalled counting measure. Explain why µ is a measure.

17. Show f : Ω→ R is measurable if and only if f−1 (U) is measurable whenever U isan open set. Hint: This is pretty easy if you recall that every open set is the disjointunion of countably many connected components.

18. The smallest σ algebra on R which contains the open intervals, denoted by B iscalled the Borel sets. Show that B contains all open sets and is also the smallest σ

algebra which contains all open sets. Show that all continuous functions g : R→ Rare B measurable. A word of advice pertaining to Borel sets: Don’t try to describea typical Borel set. Instead, use the definition that it is a set in the smallest σ algebracontaining the open sets.

19. Show that f : Ω→ R is measurable if and only if f−1 (B) is measurable for everyBorel B. Recall B is the smallest σ algebra which contains the open sets. Hint: LetG be those sets B such that f−1 (B) is measurable. Argue it is a σ algebra.

20. Now suppose f : Ω→ R where (Ω,F ) is a measureable space. Suppose g : R→ Ris B measurable. Explain why g◦ f is F measurable.

21. The open sets in Rn are defined to be all sets U which are unions of open rectanglesof the form R = ∏

ni=1 (ai,bi) Show that all open sets in Rn are a countable union of

such open rectangles. If a pi system K consists of products of open intervals like theabove, show that σ (K ) is B the Borel sets. Hint: There are countably many openrectangles of the form ∏

ni=1 (p,q) ,q, p ∈Q Show that an arbitrary open rectangle is

the union of open rectangles of this sort having rational end points.

22. ↑Show that a set of the form ∏ni=1 Bi is a Borel set in Rn if each Bi is a Borel set in R.

The Borel sets in Rn are the smallest σ algebra which contains the open sets. Hint:You might let fi :Rn→R be the projection map. Explain why f−1

i (B) is a Borel setwhen B is a Borel set in R. You know fi is continuous and that it follows that it isBorel measurable. Now consider intersections of sets like this.