Math 524
Homework due 11/29/06
Reading: Sections 3, 4 and 5, Chapter 2 in Folland.
Problems from Folland:
Chapter 2, Section 3: problem 23, 25, 26, 29 and 30.
Problem 1: Let (X , M, µ) be a measure space. Let {fn}n≥1be a sequence of non-negative
measurable functions. Assume that fn→f µ-a.e., and fn, f ∈L1(µ). Show that
lim
n→∞ Zfndµ =Zf dµ iff lim
n→∞ |fn−f|dµ = 0.
Problem 2: Lf f≥0 be a Lebesgue integrable function on [0,1]. If for every n= 1,2,· · ·
Z1
0
[f(x)]ndx =Z1
0
f(x)dx,
show that there exists a Lebesgue measurable set A∈[0,1] such that f=χAm-a.e..
Problem 3: Suppose (X, M, µ) is a measure space, and Fis a family of non-negative
integrable functions on Xwith the following properties:
1. If ϕ, ψ ∈ F, then ϕ+ψ∈ F.
2. If ϕ, ψ ∈ F, then max{ϕ, ψ} ∈ F .
3. If fis measurable, f≥0 and Rf dµ > 0, then there is ϕ∈ F such that ϕ≤fand
Rϕ dµ > 0.
Prove that if fis non-negative and measurable then
Zf dµ = sup{Zϕ dµ :ϕ∈ F, ϕ ≤f}
1