Problem 3. Let Cξ and Cν be two Cantorsets (constructed in previous HW ). Show that there exist afunction 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 acontinuous function Φ so that f ◦ Φ is non-measurable. One may usefunction F constructed above (BUT YOU NEED TO PLAY WITH IT). One ofthe ideas is to take two measurable sets C1 andC2 such that m(C1) > 0 butm(C2) = 0 and function Φ : C1 →C2, continuous. Also take N ⊂ C1 -non-measurable set and define f = χΦ(N) .
(d) Use the above construction to show that there exists aLebesque measurable set that is not a Borel set.
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THIS WAS THE PREVIUS HW, WHERE WE DEFINE Cξ andCν
Problem 3. Consider the unit interval [0, 1], and let ξ be fixedreal number with ξ ∈ (0, 1) (note that the case ξ = 1/3 correspondsto the regular Cantor set we learned in our lectures). In stage 1of the construction, remove the centrally situated open interval in[0, 1] of length ξ. In stage 2 remove the centrally situated openintervals each of relative length ξ (i.e. if the interval haslength a you remove an interval of length ξ × a), one in each ofthe remaining intervals after stage 1, and so on. Let Cξdenote the set which remains after applying the above procedureindefinitely
(a) Prove that Cξ is compact.
(b) Prove that Cξ is totally disconnected andperfect.
(c) Atually, prove that the complement of Cξ in [0,1] is the union of open intervals of total length equal to 1.
