Question

Problem 3. Let C_ξ and C_ν 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 C_ξ 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) .

(d) Use the above construction to show that there exists a Lebesque measurable set that is not a Borel set.

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THIS WAS THE PREVIUS HW, WHERE WE DEFINE C_ξ and C_ν

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with ξ ∈ (0, 1) (note that the case ξ = 1/3 corresponds to the regular Cantor set we learned in our lectures). In stage 1 of the construction, remove the centrally situated open interval in [0, 1] of length ξ. In stage 2 remove the centrally situated open intervals each of relative length ξ (i.e. if the interval has length a you remove an interval of length ξ × a), one in each of the remaining intervals after stage 1, and so on. Let C_ξ denote the set which remains after applying the above procedure indefinitely

(a) Prove that C_ξ is compact.

(b) Prove that C_ξ is totally disconnected and perfect.

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

Answer 1