Measure, Integration & Real Analysis, 2021a

48 Chapter 2 Measures
Thus
|A ∪ G| ≥ ∣ ∣A ∪ ( m ⋃
= |A| +
n=1
I n
)∣ ∣
m
∑ l(I n ).
n=1
∞
|A ∪ G| ≥|A| + ∑ l(I n )
n=1
≥|A| + |G|,
completing the proof that |A ∪ G| = |A| + |G|.
The next result shows that the outer measure of the disjoint union of two sets is
what we expect if at least one of the two sets is closed.
2.63 additivity of outer measure if one of the sets is closed
Suppose A and F are disjoint subsets of R and F is closed. Then
|A ∪ F| = |A| + |F|.
Proof Suppose I 1 , I 2 ,...is a sequence of open intervals whose union contains A ∪ F.
Let G = ⋃ ∞
k=1
I k . Thus G is an open set with A ∪ F ⊂ G. Hence A ⊂ G \ F, which
implies that
2.64 |A| ≤|G \ F|.
Because G \ F = G ∩ (R \ F), we know that G \ F is an open set. Hence we can
apply 2.62 to the disjoint union G = F ∪ (G \ F), getting
|G| = |F| + |G \ F|.
Adding |F| to both sides of 2.64 and then using the equation above gives
|A| + |F| ≤|G|
∞
≤ ∑ l(I k ).
k=1
Thus |A| + |F| ≤|A ∪ F|, which implies that |A| + |F| = |A ∪ F|.
Recall that the collection of Borel sets is the smallest σ-algebra on R that contains
all open subsets of R. The next result provides an extremely useful tool for
approximating a Borel set by a closed set.
2.65 approximation of Borel sets from below by closed sets
Suppose B ⊂ R is a Borel set. Then for every ε > 0, there exists a closed set
F ⊂ B such that |B \ F| < ε.
Measure, Integration & Real Analysis, by Sheldon Axler