668 CHAPTER 33. THE LEBESGUE MEASURE AND INTEGRAL IN Rp

(e) Next show that for x ∈ (y,z) ,

φ (y)+λ y (x− y)≤ φ (x)≤ φ (y)+az (x− y)

for some az and for x ∈ (w,y) , there is aw such that

φ (y)+λ y (x− y)≤ φ (x)≤ φ (y)+aw (x− y) .

Thus φ is continuous. Note that there is no change if φ is convex on an openinterval.

7. ↑ Prove Jensen’s inequality. If φ : R→ R is convex, µ(Ω) = 1, and f : Ω → R is inL1(Ω), then φ(

∫Ω

f du)≤∫

Ωφ( f )dµ . Hint: Let s =

∫Ω

f dµ and use Problem 6.

8. B(p,q) =∫ 1

0 xp−1(1− x)q−1dx,Γ(p) =∫

∞

0 e−tt p−1dt for p,q > 0. The first of theseis called the beta function, while the second is the gamma function. Show a.) Γ(p+1) = pΓ(p); b.) Γ(p)Γ(q) = B(p,q)Γ(p+q).

9. If X : Ω → Rp is measurable, where (Ω,F ) is a measurable space, show that

X−1 (B) ∈ F

for every B a Borel set.

10. ↑Let (Ω,F ,P) be a probability space. This means the measure P has the prop-erty that P(Ω) = 1. A random vector is a measurable function X : Ω → Rp. Theprobability distribution measure λX is defined as follows. For E Borel, λX (E) ≡P(X ∈ E). Show this gives a probability measure on the Borel sets of Rp. Some-times this measure can be realized as an integral of the form

∫Rp f (x)dmp but in

general, this will not be the case. However, it is a perfectly good measure and all thetheory of the Lebesgue integral developed above can be used.

11. ↑In the context of the above problem, suppose E is a Borel set in Rp. Note first thatthe concept of a Borel set is not even defined on Ω. Explain the following equations:∫

XE (x)dλX = λX (E)≡ P(X ∈ E) = P(X−1 (E)

)=

∫XX−1(E) (ω)dP =

∫XE (X (ω))dP

Be sure to explain why everything makes sense and is appropriately measurable.Extend this to conclude that if f is a Borel measurable nonnegative function, then∫

f dλX =∫

f (X (ω))dP

Hopefully you will see from this that there are lots of measures which are of interestother than Lebesgue measure.

12. Show that if f is any bounded Borel measurable function, then∫f dλX =

∫f (X (ω))dP