Departments of Mathematics
Fall 2015
Montana State University
Prof. Kevin Wildrick
Measure Theory
Problem Set 5
Due Friday, October 2nd , 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 S be any collection of subsets of a set Y .
(a) Show that there is a σ-algebra ΣS in Y with the following properties:
• S ⊆ ΣS ,
• if Ξ is a σ-algebra in Y with S ⊆ Ξ, then ΣS ⊆ Ξ.
The σ-algebra ΣS is called the σ-algebra generated by S.
(b) Let X be another set, and let f : X → Y be a mapping. Show that
f −1 (ΣS ) = Σf −1 (S) .
2. Let (X, Σ, µ) be a measure space. Show that if E0 ⊆ E1 ⊆ E2 ⊆ . . . is a nested sequence
of sets in Σ, then
!
[
µ
En = lim µ(En ).
n∈N
n→∞
Formulate and prove a similar results for a nested sequence E0 ⊇ E1 ⊇ E2 ⊇ . . . of sets in
Σ.
3. (?) Exercises 3K-3N in Bartle.
4. (?) Let (X, Σ, µ) be a measure space, and let F denote the collection of all functions on
X with values in R.
• Show that the relation
f ∼ g ⇐⇒ f = g µ − a.e.
is an equivalence relation on F.
• Assume that (X, Σ, µ) has the following property: if M ∈ Σ satisfies µ(M ) = 0, then
every subset of M is in Σ. Show that if f ∈ F is measurable, then so is any function
g ∈ F that is equivalent to f .
We call ΣS the σ-algebra generated by S.
5. Let (X, Σ, µ) be a measure space, and let E ∈ Σ. Show that
ΣbE := {E ∩ F : F ∈ Σ}
is a σ-algebra on E, and that µbE : ΣbE → [0, ∞] is a measure.