212 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

book. If µ is the measure which results, show that µ ((α,β )) = F (β−)−F (α+)and µ ((α,β ]) = F (β+)−F (α+) ,µ ([α,β ]) = F (β+)−F (α−) . Here F (x+)≡limy→x,y>x F (y) , F (x−)≡ limy→x,y<x F (y) Explain why the measure µ is a regularcomplete measure. It is easy from Theorem 8.2.1.

3. Let δ z ( f ) = f (z) for f ∈Cc (R) . Describe the resulting measure for which δ z = L.

4. Let L f ≡∑∞k=1 f (k) . Show this is a positive linear functional on Cc (R) and describe

the resulting Radon measure.

5. Consider the two functionals L f ≡∫

f (x)dx and Lz f ≡∫

f (x− z)dx both definedon Cc (R). Explain, using beginning calculus, why these functionals are the same.Explain why whenever f is measurable and nonnegative,∫

f (x)dm1 (x) =∫

f (x− y)dm1 (x) .

Obtain continuity of translation of Lebesgue measure right away directly from theRiesz representation theorem. Generalize to Rp.

6. Show that Lemma 8.2.10 works for metric space, not just Rp.

7. If you have a nonempty open set V in Rp, show that there is an increasing sequenceof open sets {Wn} ,Wn ⊆Wn+1, and ∪nWn =V . Next show that you can also arrangeto have Wn compact. Hint: You might consider using dist

(x,VC

)and its properties.

8. Suppose h is continuous on an open set U . Using Problem 7, verify that h(U) is aBorel set.

9. Let N be a set of measure zero with respect to Lebesgue measure. Also let h be aLipschitz continuous function meaning that for some K, ∥h(x)−h(y)∥ ≤ K ∥x−y∥and h is defined near N. Show that h(N) also has measure zero. Follow the stepsand fill in needed details.

(a) Let ε > 0. There is V open such that mp (V )< ε and V ⊇ N.

(b) For each x∈N, there is a ball Bx centered at x with B̂x contained in V . Go aheadand let the ball be taken with respect to the norm ∥x∥ ≡max{|xi| , i≤ p}. Thusthese Bx are open cubes.

(c) You know from Problem 8 that h(Bx) is measurable. Obtain countably manydisjoint balls {Bxi}

∞

i=1 such that{

B̂xi

}covers N.

(d) Explain why h(N) is covered by{

h(B̂xi

)}. Now fill in the details of the fol-

lowing estimate. mp (h(N))≤ ∑∞i=1 mp

(h(B̂xi

))≤ ∑

∞i=1 K pmp

(B̂xi

)= ∑

∞i=1 K p5pmp (Bxi) = (5K)p

ε .

(e) Now explain why this shows that mp (h(N)) = 0. Thus Lipschitz mappingstake sets of measure zero to sets of measure zero.

10. Use this and Proposition 8.3.2 to show that if h is a Lipschitz function, then if Eis Lebesgue measurable, so is h(E). Hint: This will involve completeness of themeasure and Problem 9. You could first show that it suffices to assume that E iscontained in some ball to begin with if this would make it any easier.