MAS451/6352 Measure and Probability - Additional Exercises on
Outer Measures
1. Define functions µ∗1 to µ∗6 on P(R) by
µ∗1 (A) =
µ∗2 (A) =
µ∗3 (A)
=
0
1
if A = ∅
if A =
6 ∅
0
∞
if A = ∅
if A =
6 ∅
0
1
if A is bounded
if A is unbounded

 0
∗
1
µ4 (A) =

∞
if A = ∅
if A is non-empty and bounded
if A is unbounded
0
if A is countable
∗
µ5 (A) =
1 if A is uncountable
0
if A is countable
∗
µ6 (A) =
∞ if A is uncountable
(a) Which of the above set functions are outer measures?
(b) For each i such that µ∗i is an outer measure, determine the µ∗i measurable subsets of R.
2. Let C be a countable subset of R. Using only the definition of λ∗ , show
that λ∗ (C) = 0.
3. Let S be a set, A be a Boolean algebra of subsets of S and µ be a
finitely additive measure on (S, A). For each A ⊆ S, define
∗
µ (A) = inf
∞
X
µ(An ),
n=1
where the inf isStaken over all sequences of subsets of S, (An , n ∈ N)
for which A ⊆ ∞
n=1 An .
(a) Show that µ∗ is an outer measure on S.
(b) Show that each set in A is µ∗ -measurable.
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(c) Show that if µ is countably additive, then µ∗ (A) = µ(A) for all
A ∈ A.
(d) Deduce that if µ is countably additive, then there exists a measure
on (S, σ(A)) which agrees with µ on (S, A).
4. Let c ∈ R and for all A ∈ P(R) define the set,
A + c = {x + c; x ∈ A}.
(a) Show that if A = (a, b) then A + c = (a + c, b + c).
S
S∞
(b) Deduce that if A = ∞
n=1 (an , bn ) then A + c =
n=1 (an + c, bn + c).
(c) (Translation Invariance of Lebesgue Outer Measure.) Show that
λ∗ (A + c) = λ∗ (A), for all A ⊆ R.
(d) (Translation Invariance of Lebesgue Measure.) If A ∈ B(R), show
that A + c ∈ B(R), and hence deduce that λ(A + c) = λ(A), for
all A ∈ B(R).
[Hint: Its probably best to do part (d) after Chapter 2. You may
also find it useful to introduce the translation mapping τc : R → R
defined by τc (x) = x − c, for all x ∈ R. Is τc measurable?]
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