Measure, Integration & Real Analysis, 2021a

46 Chapter 2 Measures
5 Suppose (X, S, μ) is a measure space such that μ(X) < ∞. Prove that if A is
a set of disjoint sets in S such that μ(A) > 0 for every A ∈A, then A is a
countable set.
6 Find all c ∈ [3, ∞) such that there exists a measure space (X, S, μ) with
{μ(E) : E ∈S}=[0, 1] ∪ [3, c].
7 Give an example of a measure space (X, S, μ) such that
{μ(E) : E ∈S}=[0, 1] ∪ [3, ∞].
8 Give an example of a set X,aσ-algebra S of subsets of X, a set A of subsets of X
such that the smallest σ-algebra on X containing A is S, and two measures μ and
ν on (X, S) such that μ(A) =ν(A) for all A ∈Aand μ(X) =ν(X) < ∞,
but μ ̸= ν.
9 Suppose μ and ν are measures on a measurable space (X, S). Prove that μ + ν
is a measure on (X, S). [Here μ + ν is the usual sum of two functions: if E ∈S,
then (μ + ν)(E) =μ(E)+ν(E).]
10 Give an example of a measure space (X, S, μ) and a decreasing sequence
E 1 ⊃ E 2 ⊃··· of sets in S such that
( ⋂ ∞
μ
k=1
E k
)
̸= lim
k→∞
μ(E k ).
11 Suppose (X, S, μ) is a measure space and C, D, E ∈Sare such that
μ(C ∩ D) < ∞, μ(C ∩ E) < ∞, μ(D ∩ E) < ∞.
Find and prove a formula for μ(C ∪ D ∪ E) in terms of μ(C), μ(D), μ(E),
μ(C ∩ D), μ(C ∩ E), μ(D ∩ E), and μ(C ∩ D ∩ E).
12 Suppose X is a set and S is the σ-algebra of all subsets E of X such that E is
countable or X \ E is countable. Give a complete description of the set of all
measures on (X, S).
Measure, Integration & Real Analysis, by Sheldon Axler