Departments of Mathematics
Fall 2015
Montana State University
Prof. Kevin Wildrick
Measure Theory
Problem Set 12
Due Monday, November 30th
th ,
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. (?) Consider the Lebesgue measure space ([0, 1], L, m) and the counting measure space
([0, 1], L, card). Show that m card, but that there is no measurable function f : ([0, 1], L) →
[0, ∞] such that
Z
f d card
m(E) =
E
for all E ∈ L.
2. (?) Let (X, Σ, µ) be a finite measure space, let {Ek }nk=1 be a collection of measurable sets,
and let {ck }nk=1 be a collection of non-negative real numbers. Show that
λ(E) :=
n
X
ck µ(E ∩ Ek )
k=1
is a measure on (X, Σ) that is absolutely continuous with respect to µ, and find the RadonNikodym derivative of λ with respect to µ. Bonus: Show that if {ck } is only assumed to
be a finite subset of R, then λ is a charge satisfying |λ| << µ, and find the Radon Nikodym
derivatives of its positive and negative parts.
3. (?) Let f : [0, 1] → [0, 1] be a C 1 -smooth function with the property that f 0 (x) > 0 for
all x ∈ [0, 1]. Show that λ(E) := m(f (E)) is a measure on ([0, 1], L) that is absolutely
continuous with respect to m, and find the Radon-Nikodym derivative of λ with respect
to m.
4. (?) Let µ, ν, and λ be σ-finite measures on a σ-algebra (X, Σ). Show that if λ µ ν,
then
dλ
dλ dµ
=
·
ν − a.e..
dν
dµ dν