Departments of Mathematics
Fall 2014
Montana State University
Prof. Kevin Wildrick
Measure Theory
Problem Set 2
Due Friday, September 11th , 11:00 am.
The symbol (?) indicates that this problem can not be found directly in Bartle’s book, and must
be solved and turned in. Other problems should be solved but need not be turned in.
1. (?) Let Σ and Ξ be σ-algebras on a set X. Find the smallest σ-algebra containing both Σ
and Ξ, and the largest σ-algebra contained in both Σ and Ξ.
2. Show that there is a constant C > 0, such that for any open subset U ⊆ Rn , we may write
[
U=
Ik ,
k∈N
where each {Ik }k∈N is a collection of closed cubes with disjoint interiors satisfying
dist(Ik , U c )
≤ (vol Ik )1/n ≤ C dist(Ik , U c ).
C
Warning: this is not trivial, nor is it contained in Bartle’s book.
3. (?) Let 0 ≤ s < ∞, and let δ > 0. The s-dimensional Hausdorff δ-content, of a set E ⊆ Rn
is defined by
)
(
X
[
(diam Ek )s : Ek ⊆ Rn , diam Ek < δ, E ⊆
Ek .
Hδs (E) = inf
k∈N
k∈N
• Show that Hδs (E) is a non-increasing function of δ (possibly a constant function with
value ∞ or 0). Conclude that
Hs (E) := lim Hδs (E)
δ→0
is well defined; this is called the s-dimensional Hausdorff measure of E.
• Show that Hs is monotone and sub-additive.
• Let δ > 0. Find disjoint sets A and B in R2 that are measurable (i.e., they are
contained in L), but for which
0 < Hδ1 (A) = Hδ1 (B) = Hδ1 (A ∪ B) < ∞.
4. Show that a set E ⊆ Rn is measurable if and only if for every > 0, there is a closed set
C ⊆ E such that
m∗ (E\C) < .
5. Show that for any set E ⊆ Rn and any x ∈ Rn , the set
E ⊕ x := {y + x : y ∈ E}
satisfies m∗ (E ⊕ x) = m∗ (E), and that E ⊕ x is measurable if and only if E is measurable.