fta-section3.7
DESCRIPTION
Measure theoryTRANSCRIPT
3.7 Properties of measures
In this section we prove some standard properties of measures on
σ-algebras.
To avoid unnecessary repetition, throughout this section we
assume that we are working in a measure space (X,F , µ).
Lemma 3.14 Let A1, A2, . . . , An be pairwise disjoint sets in
F . Then
µ
(
n⋃
k=1
Ak
)
=
n∑
k=1
µ(Ak) .
48
Corollary 3.15 Let A, B ∈ F with A ⊆ B. Then
µ(B) = µ(A) + µ(B \A)
and so, in particular,
µ(B) ≥ µ(A) .
49
Lemma 3.16 Let A1, A2, A3, · · · ∈ F . Then
µ
(
∞⋃
k=1
Ak
)
≤∞∑
k=1
µ(Ak) .
50
51
Theorem 3.17 Let A1, A2, A3, · · · ∈ F with
A1 ⊆ A2 ⊆ A3 ⊆ · · · .
Then
µ
(
∞⋃
k=1
Ak
)
= limn→∞
µ(An) .
52
Corollary 3.18 Let B1, B2, B3, · · · ∈ F with
B1 ⊇ B2 ⊇ B3 ⊇ · · · .
Suppose that µ(B1) < ∞. Then
µ
(
∞⋂
k=1
Bk
)
= limn→∞
µ(Bn) .
53