1.1 Section 2.3
Problem 1.
Suppose A is measurable. Then for any >), there is an open set G = G such that A ⊂ G and λ∗(G \ A) < .
Problem 2.
Suppose A is a null set. Then for any > 0, there exists a sequence of intervals Ij such that A ⊂ S∞j=1 Ij and P∞j=1 |Ij| < . Now consider A2 = {a2 : a ∈ A}. Note that if a ∈ Ik = (x,y) for some k, then
x < a < y
Problem 3.
Suppose A is any set, B is measurable, and λ∗(A4B) = 0. Note that A4B = (A \ B) ∪ (B \ A). Also note that (A \ B) ∪ (B \ A) = ∅. So we have,
λ∗(A4B) = λ ∗ (() ∪ (B \ A))
= λ∗(A \ B) + λ∗(B \ A)
= 0
Thus, we have that λ∗(A \ B) = −λ∗(B \ A). Since outer measure is non-negative, we must have that,
λ∗(A \ B) = 0 = λ∗(B \ A)
So we have that A\B is a null set and is thus measurable. Now note that A = (A∩B)∪ (A \ B) and that (A ∩ B) ∩ (A \ B) = ∅. Since B is measurable, we have that there exists an open set G such that B ⊂ G and λ∗(G \ B) <
Problem 6.
Problem 9.
1
1.2 Section 2.4 Problem 2.
Problem 4.
1.3 Section 2.5 Problem 1.
Problem 6.
Problem 7.
Problem 10.
Problem 13.
