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An Introduction to Measure Theory

Exercise 1.4.30 (Measurable functions on atomic measurable spaces)

Let (X,ℬ) be a measurable space that is atomic ℬ = 𝒜((Aα)α∈I) for atoms (Aα)α∈I of X. Show that a function f : X → ℂ is measurable if and only if it is constant on each atom, i.e., f = ∑ ⁡ α∈Icα1Aα.

Answers

  • Suppose that f is measurable, and suppose for the sake of contradiction that f is not constant on some atom Aα,α ∈ I. Then it takes at least two distinct values cα,cα′∈ ℂ on this atom. Consider the inverses f−1(cα),f−1(cα′). By measurability assumption, both are contained in B. By construction, both are not equal to Aα, i.e., f−1(cα) ∩ Aα ⊂ Aα and f−1(cα′) ∩ Aα ⊂ Aα. By construction again, both have non-empty intersection with Aα; therefore we have constructed sets in B which are non-empty and “smaller” than the atoms themselves - a contradiction.

  • Suppose that f = ∑ ⁡ α∈Icα1Aα. Pick an arbitrary open set U ⊆ ℂ. Then U either has values which do not get mapped from X onto them, or U ∩{cα : α ∈ I}.

    U = {c ∈ U : f−1(c) = ∅}∪ {c α : α ∈ I,cα ∈ U }.

    Both are disjoint, since atoms are assumed to be non-empty. Their union gives us U since any y ∈ ℂ either does not get mapped into or belongs to the range set {f(x) : x ∈ X} = {cα : α ∈ I}. Thus,

    f−1(U) = f−1({c ∈ U : f−1(c) = ∅})∪f−1({c α : α ∈ I,cα ∈ U}) = ∅∪⋃ α∈I,cα∈Uf−1(c α) = ⋃ α∈I,cα∈UAα.

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2020-08-10 00:00
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