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Let X be a set and A a σ-algebra of subsets of X. (a) A function...

Let X be a set and A a σ-algebra of subsets of X.

(a) A function f : X → R is measurable if the set {x ∈ X : f(x) > λ} belongs to A for every real number λ. Show that this holds if and only if the set {x ∈ X : f(x) ≥ λ} belongs to A for every λ ∈ R. (b) Let f : X → R be a function.

(i) Show that if f is measurable then the function f 2 is measurable.

(ii) Give an example to show that the converse of (i) is false.

(iii) Explain how (i) is used to prove that the product of two measurable functions is measurable.

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