Measure, Integration & Real Analysis, 2021a

Section 2B Measurable Spaces and Functions 27
Now we come to some easy but important properties of σ-algebras.
2.25 σ-algebras are closed under countable intersection
Suppose S is a σ-algebra on a set X. Then
(a) X ∈S;
(b) if D, E ∈S, then D ∪ E ∈Sand D ∩ E ∈Sand D \ E ∈S;
(c) if E 1 , E 2 ,...is a sequence of elements of S, then
∞⋂
k=1
E k ∈S.
Proof Because ∅ ∈Sand X = X \ ∅, the first two bullet points in the definition
of σ-algebra (2.23) imply that X ∈S, proving (a).
Suppose D, E ∈S. Then D ∪ E is the union of the sequence D, E, ∅, ∅,...of
elements of S. Thus the third bullet point in the definition of σ-algebra (2.23) implies
that D ∪ E ∈S.
De Morgan’s Laws tell us that
X \ (D ∩ E) =(X \ D) ∪ (X \ E).
If D, E ∈S, then the right side of the equation above is in S; hence X \ (D ∩ E) ∈S;
thus the complement in X of X \ (D ∩ E) is in S; in other words, D ∩ E ∈S.
Because D \ E = D ∩ (X \ E), we see that if D, E ∈S, then D \ E ∈S,
completing the proof of (b).
Finally, suppose E 1 , E 2 ,...is a sequence of elements of S. De Morgan’s Laws
tell us that
∞⋂ ∞⋃
X \ E k = (X \ E k ).
k=1
The right side of the equation above is in S. Hence the left side is in S, which implies
that X \ (X \ ⋂ ∞
k=1
E k ) ∈S. In other words, ⋂ ∞
k=1
E k ∈S, proving (c).
The word measurable is used in the terminology below because in the next section
we introduce a size function, called a measure, defined on measurable sets.
k=1
2.26 Definition measurable space; measurable set
• A measurable space is an ordered pair (X, S), where X is a set and S is a
σ-algebra on X.
• An element of S is called an S-measurable set, or just a measurable set if S
is clear from the context.
For example, if X = R and S is the set of all subsets of R that are countable or
have a countable complement, then the set of rational numbers is S-measurable but
the set of positive real numbers is not S-measurable.
Measure, Integration & Real Analysis, by Sheldon Axler