Mathematics 2224: Lebesgue integral
Homework exercise sheet 3
Due 3:50pm, Wednesday 9th March 2011
1. Let Σ be a σ-algebra and let µ : Σ → [0, ∞] be a measure.
(a) Show that µ is increasing: if E, F ∈ Σ with E ⊆ F , then µ(E) ≤ µ(F ).
[Hint: examine F \ E = F ∩ E c .]
(b) Show that µ respects set differences of sets of finite measure: if E, F ∈ Σ with E ⊆ F
and µ(F ) < ∞, then µ(F \ E) = µ(F ) − µ(E).
S
P
∞
∞
(c) Show that µ is countably subadditive: if E1 , E2 , · · · ∈ Σ, then µ
E
j=1 j ≤
j=1 µ(Ej ).
[Hint: look at the “disjoint unions trick” in the proof of Theorem 5(a)(iii).]
2. Let Σ be a σ-algebra and let µ : Σ → [0, ∞] be a measure. Suppose that E1 , E2 , E3 , . . . are sets
in Σ. Consider the equation
!
∞
[
µ
Ej = sup µ(Ej )
j=1
j≥1
where sup µ(Ej ) is shorthand for sup{µ(Ej ) : j ∈ N}.
j≥1
(a) Give an example to show that this equation is not always satisfied.
(b) Show that if E1 ⊆ E2 ⊆ E3 ⊆ . . . then the equation is satisfied.
3. Let Σ be a σ-algebra of subsets of a set X and let µ : Σ → [0, ∞] be a measure. Fix a set E ∈ Σ,
and define µE : Σ → [0, ∞] by µE (F ) = µ(E ∩ F ) for F ∈ Σ. Prove that µE is a measure and
µE (X) = µ(E).
4. (a) Let Σ be a σ-algebra. Suppose that for each P
k ∈ N, we have a measure µk : Σ → [0, ∞]
and a number αk ∈ [0, ∞]. Show that µ(E) = ∞
k=1 αk µk (E) for E ∈ Σ defines a measure
µ : Σ → [0, ∞].
P∞ P∞
P∞
P
a
You can use the identity ∞
(
a
)
=
k=1
j=1 jk , valid for ajk ∈ [0, ∞],
k=1 jk
j=1
without proof.
(b) Let α1 , α2 , α3 , . . . ∈ [0, ∞]. Show that there is a unique measure µ : P(N) → [0, ∞] so that
µ({n}) = αn for each n ∈ N. Explain why every measure µ : P(N) → [0, ∞] arises in this
way.
(c) Find a probability measure µ : P(N) → [0, ∞] so that for ∅ =
6 E ⊆ N we have µ(E) 6= 0.
5. (a) Let E ⊆ R. What is the σ-algebra of subsets of R which is generated by {E}?
(b) True or false, and why? If S and T are collections of subsets of R, then ΣS∪T = ΣS ∪ ΣT .
6. Consider Σ = {E ⊆ R : either E is countable or E c is countable}.
(a) Prove that Σ is a σ-algebra and Σ ( B.
(b) Prove that Σ is the σ-algebra generated by {x} : x ∈ R .
(c) Find a measure λ : Σ → [0, ∞] so that the only set E ∈ Σ with λ(E) = 0 is the empty set.