348 CHAPTER 14. THE LEBESGUE INTEGRAL

Then

a(m(Nab)+ ε) > am(V )≥ a∑i

m(Ii)> ∑i

∆ f (Ii)≥∑i

∑j

∆ f(

J ji

)≥ b∑

i∑

jm(

J ji

)≥ b∑

i∑

jm(

J ji ∩Nab

)= b∑

im(Nab∩ Ii) = bm(Nab)

Since ε is arbitrary and a < b, this shows m(Nab) = 0. If Nab is not bounded, apply theabove to Nab∩ (−r,r) and conclude this has measure 0. Hence so does Nab.

The countable union of Nab for a,b positive rational numbers is an exceptional set offwhich

D+ f (x) = D+ f (x)≥ D− f (x)≥ D− f (x)≥ D+ f (x)

and so these are all equal. This shows that off a set of measure zero, the function has aderivative a.e.

14.14 Exercises1. Show carefully that if S is a set whose elements are σ algebras which are subsets of

P (Ω) , the set of all subsets of Ω, then ∩S is also a σ algebra. Now let G ⊆P (Ω)satisfy property P if G is closed with respect to complements and countable disjointunions as in Dynkin’s lemma, and contains /0 and Ω. If H ⊆ G is any set whoseelements are subsets of P (Ω) which satisfies property P, then ∩H also satisfiesproperty P. Thus there is a smallest subset of G satisfying P. Show these things.

2. Show f : (Ω,F )→ R is measurable if and only if f−1 (open) ∈F . Show that ifE is any set in B (R) , then f−1 (E) ∈ F . Thus, inverse images of Borel sets aremeasurable. Next consider f : (Ω,F )→R being measurable and g :R→R is Borelmeasurable, meaning that g−1 (open) ∈ B (R). Explain why g ◦ f is measurable.Hint: You know that (g◦ f )−1 (U) = f−1

(g−1 (U)

). For your information, it does

not work the other way around. That is, measurable composed with Borel measur-able is not necessarily measurable. In fact examples exist which show that if g ismeasurable and f is continuous, then g◦ f may fail to be measurable. This is in thechapter, but show it anyway.

3. You have two finite measures defined on B (R) µ,ν . Suppose these are equal onevery open set. Show that these must be equal on every Borel set. Hint: You shoulduse Dynkin’s lemma to show this very easily.

4. You have two measures defined on B (R) which are finite and equal on every openset. Show, using Dynkin’s lemma that these are the same on all Borel sets.

5. Let µ (E) = 1 if 0 ∈ E and µ (E) = 0 if 0 /∈ E. Show this is a measure on P (R).

6. Give an example of a measure µ and a measure space and a decreasing sequence ofmeasurable sets {Ei} such that limn→∞ µ (En) ̸= µ (∩∞

i=1Ei).

7. You have a measure space (Ω,F ,P) where P is a probability measure on F . ThusP(Ω) = 1. Then you also have a measurable function X : Ω → R, meaning that