Problem 3. Let C_ξ and C_ν 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 C_ξ surjectively onto Cν.

(d) Now give an example of a measurable function f and acontinuous function Φ so that f â—¦ 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 C_ξ andC_ν

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 C_ξdenote the set which remains after applying the above procedureindefinitely

(a) Prove that C_ξ is compact.

(b) Prove that C_ξ is totally disconnected andperfect.

(c) Atually, prove that the complement of C_ξ in [0,1] is the union of open intervals of total length equal to 1.