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Exam Measure Theory

December 16, 2014, 12.00-14.45

1. Let λ be Lebesgue measure on R, and let f : R → R be given by f (x) = |x|. Describe the measure λ ◦ f−1.

2. Let X be a set and let (Y, A) be a measurable space. Let T : X → Y be surjective. Finally, f : X → R is a function from X to R.

(a) Suppose there exists an A/B measurable function g : Y → R such that f = g ◦ T . Show that f is σ(T )/B measurable.

For the remainder of this exercise suppose that f : X → R is σ(T )/B measurable.

(b) Suppose that T (x) = T (x0). Show that for all E ∈ σ(T ) we have x ∈ E if and only if x0 ∈ E.

(c) Show that for x and x0 as in (b) we have that f (x) = f (x0).

(d) Finally show that there exists an A/B measurable function h : Y → R such that f = h ◦ T .

3. (a) Formulate the Dominated Convergence Theorem.

(b) Let, for n, m = 1, 2, . . ., an(m) and an be real numbers such that an(m) → an as m → ∞. Use the Dominated Convergence Theorem to formulate a condition under which P

n=1an(m) → P

n=1an as m → ∞.

Explain your answer.

4. Let f be a non-negative measurable function on a sigma-finite measure space (X, F , µ). Let λ denote Lebesgue measure on R. Show that

Z

X

f dµ = (µ × λ) ({(x, y) ∈ X × R; 0 ≤ y ≤ f (x)}) .

Do this by first showing this is true when f is an indicator function, then for f a simple function, and finally for f a non-negative function.

5. Let f1, f2, . . . be measurable functions on a sigma-finite measure space (X, A, µ). Consider the following theorem: If P

n=1

R

X|fn|dµ < ∞, then P

n=1fn converges almost everywhere andR

X

P

n=1fndµ =P n=1

R

Xfndµ.

Prove this by using Fubini’s theorem on X × {1, 2, . . .}.

6. Let λ be Lebesgue measure on R and define on B the function µ(A) as the number of integers contained in A. Which of the following two statements is (are) true: (1) λ  µ; (2) µ  λ. Motivate your answer.

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