Real Analysis - Exercise sheet 4
November 2025
1. Consider the set X = {0, 1}N of infinite binary sequences, i.e.,
X = {α : N → {0, 1}}.
Given a finite (non-empty!) sequence of bits s = (s1 , s2 , . . . , sN ), we define its length to be |s| := N and
the associated cylindrical set defined by s to be:
Cs := {α ∈ X | α(j) = sj ∀1 ≤ j ≤ N }.
In addition, we formally define ∅ to be a cylindrical set.
(a) Show that the collection R ⊆ 2X of all cylindrical sets is a semiring.
(b) Consider the function µ0 : R → [0, ∞] defined by
µ0 (Cs ) = 2−|s|
and µ0 (∅) = 0. Show that µ0 is an elementary measure on R.
(c) Show that there is a unique measure µ on σ(R) that extends µ0 .
(d) Show that for every α ∈ X, {α} ∈ σ(R) and µ({α}) = 0.
2. Let 2 < r and let λ be the Lebesgue measure on R. Consider the set E ⊆ R defined by
cx
k
E := x ∈ R | ∃cx > 0 s.t. ∀n ∈ N, ∀k ∈ Z r ≤ x −
.
n
n
Show that E is Borel measurable and that λ(R\E) = 0.
3. Let (X, A, µ) be a measure space. We define µ∗ : 2X → [0, ∞], the inner measure induced by µ, as follows:
∀A ⊆ X,
µ∗ (A) = sup{µ(B) | B ⊆ A, B ∈ A}.
Show that if E ⊆ X is such that µ∗ (E) < ∞, then E is µ∗ - measurable if and only if µ∗ (E) = µ∗ (E).
Extra problems (not for submission)
4. Let (Rd , B(Rd ), µ) be a finite measure space. Show that for every Borel set A ⊆ Rd
µ(A) = inf µ(U ) | A ⊆ U, U ⊆ Rd open
µ(A) = sup µ(K) | K ⊆ A, K ⊆ Rd closed .
Deduce that for every Borel set A ⊆ Rd
µ(A) = sup µ(K) | K ⊆ A, K ⊆ Rd compact .
Hint: Consider the collection of all Borel sets in Rd which satisfy the first two equations above.
5. Let X = {a, b}, let E = {∅, {a}} and let µ0 be a finite elementary measure on E. Show that for every
α ∈ [0, ∞], the formulas
να (∅) = 0,
να ({b}) = α,
να ({a}) = µ0 ({a}),
να (X) = α + µ0 ({a})
define a measure να on P(X), which is an extension of µ0 . For which α is να the Carathéodory extension
of µ0 ? Why do the finite measures ν1 and ν2 which coincide on E fail to coincide on σ(E)?
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6. Let (X, M, µ) be a finite measure space and let µ∗ be the outer measure induced by µ. Suppose E ⊆ X
satisfies µ∗ (E) = µ(X) (we do not assume that E ∈ M).
(a) Show that if A, B ∈ M and A ∩ E = B ∩ E then µ(A) = µ(B).
(b) Let ME = {A ∩ E | A ∈ M} and define the function ν : ME → [0, ∞] by
∀A ∈ M,
ν(A ∩ E) = µ(A).
Show that ME is a σ-algebra on X and that ν is a measure on ME .
7. Let f : R → R be differentiable with a continuous derivative. Show that the graph of f in the plane has
planar Lebesgue measure zero.
8. Let λ be the Lebesgue measure on R and let E be a Lebesgue measurable set such that 0 < λ(E). Show
that for every 0 < t < λ(E) there exists a bounded Lebesgue measurable set A such that λ(A) = t.
9. Show that if the Lebesgue measure of a set A ⊆ R is greater than 1, then there exist distinct points x, y ∈ A
such that x − y ∈ Z.
10. Show that if a set A ⊆ R has positive Lebesgue measure, then 0 is an interior point of the set A − A :=
{x − y | x, y ∈ A}.
11. Let R be a semiring on a set X and let m be a σ-finite elementary measure on R. Suppose that (X, Σ, µ)
is the Carathéodory completion and (X, F, ν) is some completed measure space such that R ⊆ F and
ν|R = m. Show (directly) that Σ ⊆ F.
12. Let X be an uncountable
set and let D ⊆ be some countable subset with infinite elements and consider the
P
measure ν = a∈D δa (i.e., ν is an infinite sum of Dirac delta measures). Consider the semiring R ⊆ 2X
of the countable subsets and let µ0 be the restriction of ν to R. Show that ν is indeed a measure on 2X
and that µ, the Carathéodory extension of ν0 , is defined on 2X as well, but µ(A) = ∞ for all uncountable
subsets of X.
13. Let A be the collection of all finite unions of subsets of Q of the form (a, b] ∩ Q with −∞ ≤ a < b ≤ ∞.
(a) Show that A is an algebra on Q.
(b) Show that the σ-algebra generated by A on Q is 2Q .
(c) Define a function µ0 : A → [0, ∞] by µ0 (∅) = 0 and µ0 (A) = ∞ for all non empty A ∈ 2Q . Show that
µ0 is an elementary measure on A, and that there is more than one measure on 2Q whose restriction
to A is µ0 .
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