Departments of Mathematics
Fall 2014
Montana State University
Prof. Kevin Wildrick
Measure Theory
Problem Set 7
Due Friday, October 15th , 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 f , g : (X, Σ, µ) → [0, ∞] be measurable functions such that f ≤ g µ-almost everywhere, and assume
Z
g dµ < ∞.
X
Is it true that
Z
Z
Z
g dµ −
X
g − f dµ?
f dµ =
X
X
2. (?) Exercise 3.V in Bartle.
3. (?) Let {fn : (X, Σ, µ) → [0, ∞]}n∈N be a sequence of measurable functions that satisfies
f1 ≥ f2 ≥ f3 ≥ . . .
µ − a.e.
and converges µ-a.e. to a measurable function f : (X, Σ, µ) → [0, ∞]. Show that if
∞, then
Z
Z
lim
fn dµ =
f dµ.
n→∞ X
R
X
f1 dµ <
X
4. (?) 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 for every > 0, there is a number δ > 0 such that if E ∈ Σ satisfies µ(E) < δ,
then
Z
f dµ < .
E
Note: The above exercise implies that for every > 0, there is δ > 0 such that if E1 , . . . , EN
is a collection of disjoint sets in Σ satisfying
N
X
µ(Ei ) < δ,
i=1
then
N Z
X
i=1
f dµ < .
Ei
This property is often called “the absolute continuity of the integral”, in analogy to the
absolute continuity of a function.