26.8. EXERCISES 971

10. A random variable, X is a measurable real valued function defined on a measurespace, (Ω,S ,P) where P is just a measure with P(Ω) = 1 called a probabilitymeasure. The distribution function for X is the function, F (x) ≡ P([X ≤ x]) inwords, F (x) is the probability that X has values no larger than x. Show that F isa right continuous increasing function with the property that limx→−∞ F (x) = 0 andlimx→∞ F (x) = 1.

11. Suppose F is an increasing right continuous function.

(a) Show that L f ≡∫ b

a f dF is a well defined positive linear functional on Cc (R)where here [a,b] is a closed interval containing the support of f ∈Cc (R) .

(b) Using the Riesz representation theorem for positive linear functionals whichare defined on Cc (R) , let µ denote the Radon measure determined by L. Showthat µ ((a,b]) = F (b)−F (a) and µ ({b}) = F (b)−F (b−) where F (b−) ≡limx→b−F (x) .

(c) Review Corollary 20.1.4 on Page 601 at this point. Show that the conditionsof this corollary hold for µ and m. Consider µ⊥+ µ ||, the Lebesgue decom-position of µ where µ ||≪ m and there exists a set of m measure zero, N suchthat µ⊥ (E) = µ⊥ (E ∩N) . Show µ ((0,x]) = µ⊥ ((0,x])+

∫ x0 h(t)dt for some

h ∈ L1 (m) . Using Theorem 26.7.7 show h(x) = F ′ (x) m a.e. Explain whyF (x) = F (0)+S (x)+

∫ x0 F ′ (t)dt for some function, S (x) which is increasing

but has S′ (x) = 0 a.e. Note this shows in particular that a right continuousincreasing function has a derivative a.e.

12. Suppose now that G is just an increasing function defined on R. Show that G′ (x)exists a.e. Hint: You can mimic the proof of Theorem 26.7.4. The Dini derivates aredefined as

D+G(x) ≡ lim infh→0+

G(x+h)−G(x)h

,

D+G(x) ≡ lim suph→0+

G(x+h)−G(x)h

D−G(x) ≡ lim infh→0+

G(x)−G(x−h)h

,

D−G(x) ≡ lim suph→0+

G(x)−G(x−h)h

.

When D+G(x) = D+G(x) the derivative from the right exists and when D−G(x) =D−G(x) , then the derivative from the left exists. Let (a,b) be an open interval andlet

Npq ≡{

x ∈ (a,b) : D+G(x)> q > p > D+G(x)}.

Let V ⊆ (a,b) be an open set containing Npq such that m(V ) < m(Npq)+ ε . Showusing a Vitali covering theorem there is a disjoint sequence of intervals contained in