7-10

(a) Show that if f is absolutely continuous on [a, b] and Z is a subset of [a, b] of measure zero, then the image set defined by f(Z) = {w : w = f(z), z ∈ Z} also has measure zero. Deduce that the image under f of any measurable subset of [a, b] is measurable. (Compare Theorem 3.33.) (Hint: use the fact that the image of an interval [ai, bi] is an interval of length at most V (bi) − V (ai).)
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