Measure, Integration & Real Analysis, 2021a

72 Chapter 2 Measures
9 Suppose F 1 ,...,F n are disjoint closed subsets of R. Prove that if
g : F 1 ∪···∪F n → R
is a function such that g| Fk is a continuous function for each k ∈{1,...,n},
then g is a continuous function.
10 Suppose F ⊂ R is such that every continuous function from F to R can be
extended to a continuous function from R to R. Prove that F is a closed subset
of R.
11 Prove or give a counterexample: If F ⊂ R is such that every bounded continuous
function from F to R can be extended to a continuous function from R to R,
then F is a closed subset of R.
12 Give an example of a Borel measurable function f from R to R such that there
does not exist a set B ⊂ R such that |R \ B| = 0 and f | B is a continuous
function on B.
13 Prove or give a counterexample: If f t : R → R is a Borel measurable function
for each t ∈ R and f : R → (−∞, ∞] is defined by
then f is a Borel measurable function.
f (x) =sup{ f t (x) : t ∈ R},
14 Suppose b 1 , b 2 ,...is a sequence of real numbers. Define f : R → [0, ∞] by
⎧ ∞
1 ⎪⎨ ∑
f (x) = k=1
4 k if x /∈ {b 1 , b 2 ,...},
|x − b k |
⎪⎩
∞ if x ∈{b 1 , b 2 ,...}.
Prove that |{x ∈ R : f (x) < 1}| = ∞.
[This exercise is a variation of a problem originally considered by Borel. If
b 1 , b 2 ,... contains all the rational numbers, then it is not even obvious that
{x ∈ R : f (x) < ∞} ̸= ∅.]
15 Suppose B is a Borel set and f : B → R is a Lebesgue measurable function.
Show that there exists a Borel measurable function g : B → R such that
|{x ∈ B : g(x) ̸= f (x)}| = 0.
Measure, Integration & Real Analysis, by Sheldon Axler