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Let (X, B, µ) be a σ-finite measure space and f a nonnegative mea- surable function on X

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Real Analysis Homework #8

Due 11/24 1. Let (X, B, µ) be a σ-finite measure space and f a nonnegative mea- surable function on X. Prove that for λ := Lebesgue measure, R f dµ = (µ × λ){(x, y) : 0 < y < f (x)} (”the integral is the area under the curve”).

2. Let (X, B, µ) be σ-finite and f any measurable real-valued function on X.

Prove that (µ × λ){(x, y) : y = f (x)} = 0 (the graph of a real measurable function has measure zero).

3. Let (Ω, B, µ) be a σ-finite measure space. Assume that φ : [0, ∞) → [0, ∞) is strictly increasing and φ(0) = 0. show that

Z

Ω

φ(f (x))dµ = Z ∞

0

φ0(t)µ({x : f (x) > t})dt.

4. For I := [0, 1] with Borel σ-algebra and Lebesgue measure λ, take the cube I3 with product measure (volume) λ3 = λ × λ × λ. Let f (x, y, z) :=

1/p|y − z| for y 6= z, f(x, y, z) := +∞ for y = z. Show that f is integrable for λ3, but that for each z ∈ I, the set of y such thatR f (x, y, z)dλ(x) = +∞

is non-empty and depends on z.

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