Measure, Integration & Real Analysis, 2021a

Section 2D Lebesgue Measure 61
11 Prove that if A ⊂ R and |A| > 0, then there exists a subset of A that is not
Lebesgue measurable.
12 Suppose b < c and A ⊂ (b, c). Prove that A is Lebesgue measurable if and
only if |A| + |(b, c) \ A| = c − b.
13 Suppose A ⊂ R. Prove that A is Lebesgue measurable if and only if
for every n ∈ Z + .
|(−n, n) ∩ A| + |(−n, n) \ A| = 2n
14 Show that 1 4 and 9 13
are both in the Cantor set.
15 Show that 13
17
is not in the Cantor set.
16 List the eight open intervals whose union is G 4 in the definition of the Cantor
set (2.74).
17 Let C denote the Cantor set. Prove that 1 2 C + 1 2
C =[0, 1].
18 Prove that every open interval of R contains either infinitely many or no elements
in the Cantor set.
19 Evaluate
∫ 1
0
Λ, where Λ is the Cantor function.
20 Evaluate each of the following:
(a) Λ ( )
9
13
;
(b) Λ(0.93).
21 Find each of the following sets:
(a) Λ −1( { 1 3 }) ;
(b) Λ −1( {
16 5 }) .
22 (a) Suppose x is a rational number in [0, 1]. Explain why Λ(x) is rational.
(b) Suppose x ∈ C is such that Λ(x) is rational. Explain why x is rational.
23 Show that there exists a function f : R → R such that the image under f of
every nonempty open interval is R.
24 For A ⊂ R, the quantity
sup{|F| : F is a closed bounded subset of R and F ⊂ A}
is called the inner measure of A.
(a) Show that if A is a Lebesgue measurable subset of R, then the inner measure
of A equals the outer measure of A.
(b) Show that inner measure is not a measure on the σ-algebra of all subsets
of R.
Measure, Integration & Real Analysis, by Sheldon Axler