Measure, Integration & Real Analysis, 2021a

44 Chapter 2 Measures
For convenience, let E 0 = ∅. Then
∞⋃
k=1
E k =
∞⋃
(E j \ E j−1 ),
j=1
where the union on the right side is a disjoint union. Thus
( ⋃ ∞
μ
k=1
E k
)
=
∞
∑ μ(E j \ E j−1 )
j=1
k
= lim
k→∞
∑ μ(E j \ E j−1 )
j=1
= lim
k→∞
k
∑
j=1
(
μ(Ej ) − μ(E j−1 ) )
as desired.
= lim
k→∞
μ(E k ),
Another mew.
Measures also behave well with respect to decreasing intersections (but see Exercise
10, which shows that the hypothesis μ(E 1 ) < ∞ below cannot be deleted).
2.60 measure of a decreasing intersection
Suppose (X, S, μ) is a measure space and E 1 ⊃ E 2 ⊃··· is a decreasing
sequence of sets in S, with μ(E 1 ) < ∞. Then
Proof
( ⋂ ∞
μ
k=1
E k
)
= lim
k→∞
μ(E k ).
One of De Morgan’s Laws tells us that
∞⋂ ∞⋃
E 1 \ E k = (E 1 \ E k ).
k=1
Now E 1 \ E 1 ⊂ E 1 \ E 2 ⊂ E 1 \ E 3 ⊂···is an increasing sequence of sets in S.
Thus 2.59, applied to the equation above, implies that
(
μ E 1 \
∞⋂
k=1
Use 2.57(b) to rewrite the equation above as
( ⋂ ∞
μ(E 1 ) − μ
which implies our desired result.
k=1
k=1
E k
)
= lim
k→∞
μ(E 1 \ E k ).
E k
)
= μ(E 1 ) − lim
k→∞
μ(E k ),
Measure, Integration & Real Analysis, by Sheldon Axler