Measure, Integration & Real Analysis, 2021a

60 Chapter 2 Measures
EXERCISES 2D
1 (a) Show that the set consisting of those numbers in (0, 1) that have a decimal
expansion containing one hundred consecutive 4s is a Borel subset of R.
(b) What is the Lebesgue measure of the set in part (a)?
2 Prove that there exists a bounded set A ⊂ R such that |F| ≤|A|−1 for every
closed set F ⊂ A.
3 Prove that there exists a set A ⊂ R such that |G \ A| = ∞ for every open set G
that contains A.
4 The phrase nontrivial interval is used to denote an interval of R that contains
more than one element. Recall that an interval might be open, closed, or neither.
(a) Prove that the union of each collection of nontrivial intervals of R is the
union of a countable subset of that collection.
(b) Prove that the union of each collection of nontrivial intervals of R is a Borel
set.
(c) Prove that there exists a collection of closed intervals of R whose union is
not a Borel set.
5 Prove that if A ⊂ R is Lebesgue measurable, then there exists an increasing
sequence F 1 ⊂ F 2 ⊂··· of closed sets contained in A such that
∣
∣A \
∞⋃
k=1
F k
∣ ∣∣ = 0.
6 Suppose A ⊂ R and |A| < ∞. Prove that A is Lebesgue measurable if and
only if for every ε > 0 there exists a set G that is the union of finitely many
disjoint bounded open intervals such that |A \ G| + |G \ A| < ε.
7 Prove that if A ⊂ R is Lebesgue measurable, then there exists a decreasing
sequence G 1 ⊃ G 2 ⊃··· of open sets containing A such that
( ⋂ ∞ ) ∣ G k \ A∣ = 0.
k=1
8 Prove that the collection of Lebesgue measurable subsets of R is translation
invariant. More precisely, prove that if A ⊂ R is Lebesgue measurable and
t ∈ R, then t + A is Lebesgue measurable.
9 Prove that the collection of Lebesgue measurable subsets of R is dilation invariant.
More precisely, prove that if A ⊂ R is Lebesgue measurable and t ∈ R,
then tA (which is defined to be {ta : a ∈ A}) is Lebesgue measurable.
10 Prove that if A and B are disjoint subsets of R and B is Lebesgue measurable,
then |A ∪ B| = |A| + |B|.
Measure, Integration & Real Analysis, by Sheldon Axler