Measure, Integration & Real Analysis, 2021a

34 Chapter 2 Measures
2.42 Definition increasing function; strictly increasing
Suppose X ⊂ R and f : X → R is a function.
• f is called increasing if f (x) ≤ f (y) for all x, y ∈ X with x < y.
• f is called strictly increasing if f (x) < f (y) for all x, y ∈ X with x < y.
2.43 every increasing function is Borel measurable
Every increasing 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 increasing. To prove that f
is Borel measurable, fix a ∈ R.
Let b = inf f −1( (a, ∞) ) . Then it is easy to see that
f −1( (a, ∞) ) =(b, ∞) ∩ X or f −1( (a, ∞) ) =[b, ∞) ∩ X.
Either way, we can conclude that f −1( (a, ∞) ) is a Borel set.
Now 2.39 implies that f is a Borel measurable function.
The next result shows that measurability interacts well with composition.
2.44 composition of measurable functions
Suppose (X, S) is a measurable space and f : X → R is an S-measurable
function. Suppose g is a real-valued Borel measurable function defined on a
subset of R that includes the range of f . Then g ◦ f : X → R is an S-measurable
function.
Proof Suppose B ⊂ R is a Borel set. Then (see 2.34)
(g ◦ f ) −1 (B) = f −1( g −1 (B) ) .
Because g is a Borel measurable function, g −1 (B) is a Borel subset of R. Because f
is an S-measurable function, f −1( g −1 (B) ) ∈S. Thus the equation above implies
that (g ◦ f ) −1 (B) ∈S. Thus g ◦ f is an S-measurable function.
2.45 Example if f is measurable, then so are − f , 1 2 f , | f |, f 2
Suppose (X, S) is a measurable space and f : X → R is S-measurable. Then 2.44
implies that the functions − f , 1 2 f , | f |, f 2 are all S-measurable functions because
each of these functions can be written as the composition of f with a continuous (and
thus Borel measurable) function g.
Specifically, take g(x) =−x, then g(x) = 1 2
x, then g(x) =|x|, and then
g(x) =x 2 .
Measure, Integration & Real Analysis, by Sheldon Axler