Homework 06, Math 545 23 September, 2015

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1. Let (X, A, µ) be a space. Suppose that s : X → R is a measurable n with the property that Range(s)= {ai }i=1, where the ai ∈ R are distinct. Clearly s is a (measurable) simple function, as it can be written as

n X s = ai χAi ,Ai = {x ∈ X | s(x) = ai } ∈ A, i=1

n where X = ∪i=1Ai and Ai ∩ Aj = ∅, for i 6= j. This is called the canonical form of s. (a) Suppose that t : X → R is a measurable simple function, in the sense that it may be written as m X t = bi χBi , i=1

where Bi ∈ A, for each i = 1, ··· , m, but the bi are not necessarily distinct, the Bi are not necessarily pairwise disjoint and, finally it is not necessarily true m that X = ∪i=1Bi . (We use this second definition as the definition of a general measurable simple function.) Show that t has a finite range, and, therefore, it can be written in canonical form. (b) Suppose that s, t : X → R are a measurable simple functions, not necessarily in canonical form: n n X X s = ai χAi , t = bj χBj . i=1 j=1 Prove that, if s = t on X, then

m n X X I(s) = ai µ(Ai ) = bj µ(Bj ) = I(t). i=1 j=1 This number I( · ) we will call the simple function . (c) Suppose that f : X → R is a non-negative µ-. Define

S(f ) := {s : X → R | s is a µ-measurable simple function and 0 ≤ s ≤ f } , and recall that Z f dµ := sup {I(s) | s ∈ S(f )} .

1 If t : X → R is a non-negative µ-measurable simple function, prove that R tdµ = I(t). 2. Let (X, A, µ) be a measure space, and suppose that f : X → R is a non-negative µ-measurable function. Prove that Z Z min(f , n)dµ → f dµ, as n → ∞.

Do not use the convergence theorems of chapter 7.

3. Let (X, A, µ) be a measure space, with µ(X) < ∞, and suppose that fn : X → R is a sequence of bounded µ-measurable functions that converge to a function f uniformly. Prove that Z Z lim fndµ = f dµ. n→∞ Do not use the convergence theorems of chapter 7.

4. Let (X, A, µ) be a measure space, and suppose that fn : X → R is a sequence of non-negative, µ-measurable, and integrable functions. Suppose that fn → f point- R R wise a.e., such that f (x) is finite for each x ∈ X, fndµ → f dµ, as n → ∞, and f is integrable. Prove that, for each A ∈ A, Z Z fndµ → f dµ, as n → ∞. A A

5. Let (X, A, µ) be a measure space, and suppose that f , fn : X → R are µ-measurable R R and integrable functions, fn → f point-wise a.e., and |fn|dµ → |f |dµ. Prove that, as n → ∞, Z |fn − f |dµ → 0.

Hint: Use the result of Exercise 7.2 from the book. 6. Suppose that f : R → R is Lebesgue integrable, a ∈ R and Z x F (x) := f (s)ds. a

Prove that F :[a, ∞) → R is continuous.

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