Departments of Mathematics
Fall 2014
Montana State University
Prof. Kevin Wildrick
Measure Theory
Problem Set 6
Due Friday, October 9th , 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. (?) Let (X, Σ, µ) be a measure space, and let φ : X → [0, ∞) be a simple function with
standard representation
n
X
φ=
a i χA i .
i=1
Show that φ is measurable if and only if for every i = 1, . . . , n, the set Ai is measurable.
2. Exercise 4A in Bartle.
3. (?) Exercise 4L in Bartle.
4. (?) Exercise 4Q in Bartle.
5. (?) Let (X, Σ, µ) be a measure space, and let f : X → [0, ∞] be a measurable function
such that
Z
f dµ < ∞.
X
Show that there is a constant 0 ≤ C < ∞ such that for each λ ∈ [0, ∞],
µ({x ∈ X : f (x) > λ} ≤
C
.
λ