Departments of Mathematics
Fall 2015
Montana State University
Prof. Kevin Wildrick
Measure Theory
Problem Set 8
Due Friday, October 30th , 11:00 am.
The symbol (?) indicates that this problem must be solved and turned in. Other problems
should be solved but need not be turned in.
1. (?) Exercise 5.K in Bartle
2. (?) Exercise 5.L in Bartle.
3. Read pages 60 and 61 in Bartle.
4. (?) Let (X, Σ, µ) be a measure space with µ(X) < ∞.
a) Show that if 1 ≤ p ≤ p0 ≤ ∞ and f ∈ Lp0 (X, Σ, µ), then f ∈ Lp (X, Σ, µ) and
1
||f ||p ≤ µ(X) p
− p10
||f ||p0 .
(1)
b) For a set E ∈ Σ with µ(E) 6= 0, and a measurable function g : (X, σ, µ) → R denote
Z
Z
1
g dµ,
− g dµ :=
µ(E) E
E
whenever the integral exists. Show that under the assumptions of part a),
Z
1 Z
10
p
p
p
p0
.
− |f | dµ
≤ − |f | dµ
E
E
5. (?) Let X = N, let Σ be the collection of all subsets of N, and µ be the counting measure.
Describe Lp (X, Σ, µ) for 1 ≤ p ≤ ∞, and determine for which p and p0 it holds that
Lp0 (X, Σ, µ) ⊆ Lp (X, Σ, µ).
Is there an analogue of the inequality (1)?