MATH 507/420 - Assignment #5
Due on Friday November 25, 2011
Name —————————————–
Student number —————————
1
Problem 1:
Let (X, M, µ) be a measure space. Prove that L∞ (X, µ) is complete.
2
Problem 2:
Let
µ) be a measure space and p ∈ (0, 1). Prove that ||f ||p =
R (X,p M,1/p
( X |f | dµ) does not define a norm on Lp (X, µ).
3
Problem 3:
Prove the uniqueness assertion in the Lebesgue decomposition theorem.
4
Problem 4:
For p ∈ (1, ∞), show that if the absolutely continuous function f on
[a, b] is the indefinite integral of an Lp ([a, b]) function, then there is a
constant K such that for any partition {x0 , x1 , ..., xn } of [a, b],
n
X
|f (xk ) − f (xk−1 )|p
≤ K.
p−1
|x
k − xk−1 |
k=1
5
Problem 5:
Let (X, M) be a measurable space and µ and ν two σ-finite measures
on it. If µ ν and ν µ, show that
dν dµ
·
= 1 a.e. w.r.t. µ.
dµ dν
6
Problem 6:
Formulate and prove an extension of Hölder’s inequality for the product
of three functions.