Measure, Integration & Real Analysis, 2021a

Section 2B Measurable Spaces and Functions 33
We have been dealing with S-measurable functions from X to R in the context
of an arbitrary set X and a σ-algebra S on X. An important special case of this
setup is when X is a Borel subset of R and S is the set of Borel subsets of R that are
contained in X (see Exercise 11 for another way of thinking about this σ-algebra). In
this special case, the S-measurable functions are called Borel measurable.
2.40 Definition Borel measurable function
Suppose X ⊂ R. A function f : X → R is called Borel measurable if f −1 (B) is
a Borel set for every Borel set B ⊂ R.
If X ⊂ R and there exists a Borel measurable function f : X → R, then X must
be a Borel set [because X = f −1 (R)].
If X ⊂ R and f : X → R is a function, then f is a Borel measurable function if
and only if f −1( (a, ∞) ) is a Borel set for every a ∈ R (use 2.39).
Suppose X is a set and f : X → R is a function. The measurability of f depends
upon the choice of a σ-algebra on X. If the σ-algebra is called S, then we can discuss
whether f is an S-measurable function. If X is a Borel subset of R, then S might
be the set of Borel sets contained in X, in which case the phrase Borel measurable
means the same as S-measurable. However, whether or not S is a collection of Borel
sets, we consider inverse images of Borel subsets of R when determining whether a
function is S-measurable.
The next result states that continuity interacts well with the notion of Borel
measurability.
2.41 every continuous function is Borel measurable
Every continuous real-valued function defined on a Borel subset of R is a Borel
measurable function.
Proof Suppose X ⊂ R is a Borel set and f : X → R is continuous. To prove that f
is Borel measurable, fix a ∈ R.
If x ∈ X and f (x) > a, then (by the continuity of f ) there exists δ x > 0 such that
f (y) > a for all y ∈ (x − δ x , x + δ x ) ∩ X. Thus
f −1( (a, ∞) ) ( ⋃
)
=
x∈ f −1( ) (x − δ x, x + δ x ) ∩ X.
(a, ∞)
The union inside the large parentheses above is an open subset of R; hence its
intersection with X is a Borel set. Thus we can conclude that f −1( (a, ∞) ) is a Borel
set.
Now 2.39 implies that f is a Borel measurable function.
Next we come to another class of Borel measurable functions. A similar definition
could be made for decreasing functions, with a corresponding similar result.
Measure, Integration & Real Analysis, by Sheldon Axler