Math 318 HW #9
Due 5:00 PM Thursday, April 14
Reading:
Wilcox & Myers §28–30.
Problems:
1. Let A be a bounded, measurable set and let (fn ) be a sequence of measurable functions on A
converging to f . Suppose ϕ ∈ L(A) and that |fn (x)| ≤ ϕ(x) for all x ∈ A and all n = 1, 2, . . ..
Show that
Z
Z
lim
fn g dm =
f g dm
n→∞ A
A
if g is measurable and essentially bounded on A, meaning that there exists M > 0 such that
|g(x)| ≤ M a.e.
(Comment: If g is essentially bounded, then the quantity
(
ess sup |g(x)| :=
x∈A
inf
Z⊂A
m(Z)=0
sup |g(x)| ,
x∈A\Z
called the essential supremum of A, is finite.)
2. Exercise 31.16.
3. Exercise 31.33. You may find Problem 1(b) from HW 8 useful.
1
)