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Problem 3. Let Cξ and Cν be two Cantor sets (constructed in previous HW ). Show...

Problem 3. Let Cξ and Cν be two Cantor sets (constructed in previous HW ). Show that there exist a function F : [0, 1] → [0, 1] with the following properties

(a) F is continuous and bijective.

(b)F is monotonically increasing.

(c) F maps Cξ surjectively onto Cν.

(d) Now give an example of a measurable function f and a continuous function Φ so that f ◦ Φ is non-measurable. One may use function F constructed above (BUT YOU NEED TO PLAY WITH IT). One of the ideas is to take two measurable sets C1 and C2 such that m(C1) > 0 but m(C2) = 0 and function Φ : C1 → C2, continuous. Also take N ⊂ C1 - non-measurable set and define f = χΦ(N) .

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Colby Messinger answered 1 year ago
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