Measure, Integration & Real Analysis, 2021a

40 Chapter 2 Measures
23 Suppose f : R → R is a strictly increasing function. Prove that the inverse
function f −1 : f (R) → R is a continuous function.
[Note that this exercise does not have as a hypothesis that f is continuous.]
24 Suppose f : R → R is a strictly increasing function and B ⊂ R is a Borel set.
Prove that f (B) is a Borel set.
25 Suppose B ⊂ R and f : B → R is an increasing function. Prove that there exists
a sequence f 1 , f 2 ,...of strictly increasing functions from B to R such that
for every x ∈ B.
f (x) = lim
k→∞
f k (x)
26 Suppose B ⊂ R and f : B → R is a bounded increasing function. Prove that
there exists an increasing function g : R → R such that g(x) = f (x) for all
x ∈ B.
27 Prove or give a counterexample: If (X, S) is a measurable space and
f : X → [−∞, ∞]
is a function such that f −1( (a, ∞) ) ∈ S for every a ∈ R, then f is an
S-measurable function.
28 Suppose f : B → R is a Borel measurable function. Define g : R → R by
{
f (x) if x ∈ B,
g(x) =
0 if x ∈ R \ B.
Prove that g is a Borel measurable function.
29 Give an example of a measurable space (X, S) and a family { f t } t∈R such
that each f t is an S-measurable function from X to [0, 1], but the function
f : X → [0, 1] defined by
f (x) =sup{ f t (x) : t ∈ R}
is not S-measurable.
[Compare this exercise to 2.53, where the index set is Z + rather than R.]
30 Show that
(
lim
j→∞
lim
k→∞
( ) ) 2k cos(j!πx) =
{
1 if x is rational,
0 if x is irrational
for every x ∈ R.
[This example is due to Henri Lebesgue.]
Measure, Integration & Real Analysis, by Sheldon Axler