Assignment 8 of Math 5302
1. (a) Prove Property M6: If the Ak’s are measurable and A1 ⊇ A2 ⊇ A3 · · · , and if λ(A1) < ∞, then
λ(
\∞
k=1
Ak) = lim
k→∞
λ(Ak).
(b) Give an example to show why the assumption λ(A1) < ∞ is needed in Property M6.
2. Let A and B be subsets of R
n
.
(a) Suppose that A and B are measurable. Prove that
λ(A) + λ(B) = λ(A ∪ B) + λ(A ∩ B).
(b) Prove that in general
λ
∗
(A) + λ
∗
(B) ≥ λ
∗
(A ∪ B) + λ
∗
(A ∩ B).
3. Let a ∈ R be fixed. Prove that
λ({a} × R
n−1
) = 0.
4. (a) Let > 0. Prove that there exists an open set G ⊂ R such that Q ⊂ G and λ(G) < .
(b) Construct a closed subset of [0, 1] whose measure is positive and whose interior is empty.
(Hint: Try the complement of G in (a).)
5. Prove that if E ⊂ R
n
and λ
∗
(E) < ∞, then there exists a measurable set A such that E ⊆ A and
λ
∗
(E) = λ(A).
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