APPENDIX A
Measure and Integration
1 MEASURES AND THE CARATH
´
EODORY EXTENSION.
Let S be a nonempty set. A class A of subsets of S is a field,oranalgebra if
(i) ∅∈A, S ∈A, (ii) A ∈A=⇒ A
c
∈A, (iii) A, B ∈A=⇒ A ∪ B ∈A. Note that
(ii) and (iii) imply that A is closed under finite unions and finite intersections. If (iii)
is replaced by (iii)
: A
n
∈A(n =1, 2,...)=⇒∪
∞
n=1
A
n
∈A, then A is said to be
a σ-field,oraσ-algebra. Note that (iii)
implies (iii), and that a σ-field is closed
under countable intersections.
A function µ : A→[0, ∞] is said to be a measure on a field A if µ(∅)=0
and µ(∪
∞
n=1
A
n
)=
∞
n=1
µ(A
n
) for every sequence of pairwise disjoint sets A
n
∈A
(n =1, 2,...) such that ∪
∞
n=1
A
n
∈A. Note that this property, known as countable
additivity, implies finite additivity (by letting A
n
= ∅ for n ≥ m for some m,say).
Ameasureµ on a field A is σ-finite if there exists a sequence A
n
∈A(n =1, 2,...)
such that ∪
∞
n=1
A
n
= S and µ(A
n
) < ∞ for every n.
If µ is a measure on a field A,andA
n
∈A(n ≥ 1), A ⊆∪
n
A
n
,A ∈A, then
µ(A) ≤
∞
n=1
µ(A
n
) (subadditivity). To see this write B
1
= A
1
,B
n
= A
c
1
∩ ...∩
A
c
n−1
∩A
n
(n ≥ 2). Then B
n
(n ≥ 1) are disjoint, ∪
∞
n=1
A
n
= ∪
∞
n=1
B
n
,sothatµ(A)=
µ(A ∩ (∪
∞
n=1
B
n
)) =
∞
n=1
µ(A ∩ B
n
) ≤
∞
n=1
µ(A
n
) (since B
n
⊆ A
n
for all n).
Let µ be a measure on a σ-field F on S. Then F is said to be µ-complete if all
subsets of µ-null sets in F belong to F : N ∈F, µ(N)=0,B ⊆ N =⇒ B ∈F.
In this case the measure µ is also said to be complete. Given any measure µ on a
σ-field F, it is simple to check that the class of sets
F = {C = A ∪B : A ∈F,B⊆ N for some N ∈Fsuch that µ(N )=0} (1.1)
