Departments of Mathematics
Fall 2015
Montana State University
Prof. Kevin Wildrick
Measure Theory
Review Set 1
1. Let f : X → Y be a mapping. If Σ is a σ-algebra on Y show that f −1 (Σ) is a σ-algebra
on X. Is it true that if Σ is a σ-algebra on X, then f (Σ) is a σ-algebra on Y ?
2. Suppose that E ⊆ Rn is a measurable set, and F ⊆ Rn is any set disjoint from E. Show
that
m∗ (E ∪ F ) = m(E) + m∗ (F ).
3. Suppose that A and B are measurable subsets of [0, 1] with
m(A) = 1 = m(B).
Show that for each λ ∈ (0, 1), there are points a ∈ A and b ∈ B with λ = a − b.
4. Let E ⊆ Rn be any set and let > 0. Show that there is an open set U ⊇ E with
m(U ) ≤ m∗ (E) + .
5. Let D denote the open unit ball in R2 . Show, using only measure theory and basic
trigonometry, that for any k ∈ N
k sin(π/k) ≤ m(D) ≤ k tan(π/k).