Measure, Integration & Real Analysis, 2021a

Section 2B Measurable Spaces and Functions 37
2.51 Definition measurable function
Suppose (X, S) is a measurable space. A function f : X → [−∞, ∞] is called
S-measurable if
f −1 (B) ∈S
for every Borel set B ⊂ [−∞, ∞].
The next result, which is analogous to 2.39, states that we need not consider all
Borel subsets of [−∞, ∞] when taking inverse images to determine whether or not a
function with values in [−∞, ∞] is S-measurable.
2.52 condition for measurable function
Suppose (X, S) is a measurable space and f : X → [−∞, ∞] is a function such
that
f −1( (a, ∞] ) ∈S
for all a ∈ R. Then f is an S-measurable function.
The proof of the result above is left to the reader (also see Exercise 27 in this
section).
We end this section by showing that the pointwise infimum and pointwise supremum
of a sequence of S-measurable functions is S-measurable.
2.53 infimum and supremum of a sequence of S-measurable functions
Suppose (X, S) is a measurable space and f 1 , f 2 ,... is a sequence of
S-measurable functions from X to [−∞, ∞]. Define g, h : X → [−∞, ∞] by
g(x) =inf{ f k (x) : k ∈ Z + } and h(x) =sup{ f k (x) : k ∈ Z + }.
Then g and h are S-measurable functions.
Proof
Let a ∈ R. The definition of the supremum implies that
h −1( (a, ∞] ) =
∞⋃
k=1
f k
−1 ( (a, ∞] ) ,
as you should verify. The equation above, along with 2.52, implies that h is an
S-measurable function.
Note that
g(x) =− sup{− f k (x) : k ∈ Z + }
for all x ∈ X. Thus the result about the supremum implies that g is an S-measurable
function.
Measure, Integration & Real Analysis, by Sheldon Axler