Measure, Integration & Real Analysis, 2021a

36 Chapter 2 Measures
The next result shows that the pointwise limit of a sequence of S-measurable
functions is S-measurable. This is a highly desirable property (recall that the set of
Riemann integrable functions on some interval is not closed under taking pointwise
limits; see Example 1.17).
2.48 limit 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 R. Suppose lim k→∞ f k (x) exists for each
x ∈ X. Define f : X → R by
Then f is an S-measurable function.
f (x) = lim
k→∞
f k (x).
Proof
Suppose a ∈ R. We will show that
2.49 f −1( (a, ∞) ) =
∞⋃ ∞⋃ ∞⋂
−1
f ( k (a + 1 j , ∞)) ,
j=1 m=1 k=m
which implies that f −1( (a, ∞) ) ∈S.
To prove 2.49, first suppose x ∈ f −1( (a, ∞) ) . Thus there exists j ∈ Z + such that
f (x) > a + 1 j . The definition of limit now implies that there exists m ∈ Z+ such
that f k (x) > a + 1 j
for all k ≥ m. Thus x is in the right side of 2.49, proving that the
left side of 2.49 is contained in the right side.
To prove the inclusion in the other direction, suppose x is in the right side of 2.49.
Thus there exist j, m ∈ Z + such that f k (x) > a + 1 j
for all k ≥ m. Taking the
limit as k → ∞, we see that f (x) ≥ a + 1 j
> a. Thus x is in the left side of 2.49,
completing the proof of 2.49. Thus f is an S-measurable function.
Occasionally we need to consider functions that take values in [−∞, ∞]. For
example, even if we start with a sequence of real-valued functions in 2.53, we might
end up with functions with values in [−∞, ∞]. Thus we extend the notion of Borel
sets to subsets of [−∞, ∞], as follows.
2.50 Definition Borel subsets of [−∞, ∞]
A subset of [−∞, ∞] is called a Borel set if its intersection with R is a Borel set.
In other words, a set C ⊂ [−∞, ∞] is a Borel set if and only if there exists
a Borel set B ⊂ R such that C = B or C = B ∪{∞} or C = B ∪{−∞} or
C = B ∪{∞, −∞}.
You should verify that with the definition above, the set of Borel subsets of
[−∞, ∞] is a σ-algebra on [−∞, ∞].
Next, we extend the definition of S-measurable functions to functions taking
values in [−∞, ∞].
Measure, Integration & Real Analysis, by Sheldon Axler