Question

3. (3 pts) Let V (0,1be the Vitali set introduced in the lectures. Let u be the outer measure on Rwhose restriction to the u-measurable sets gives rise to the Lebesgue measure. Then choose the correct answer): (a) H(V) is not defined; (b) H(V) = 0; (c) H(V) >0. Justify your answer (use notations and results from the lectures): .

Answer 1

restriction to the in the outer measure on R whose le- measurable sets gives rise to the lebesque measure. V c [0, 1] be the vitali set. construction of vi classes Define a relation 'n' on [o,i ] as a neny iff x-ye G Where x, y e[0, 1] then 'n' is an equivalance relation. the set v consits exactly one members from each equivalare then for v we have two properties - (i) the difference of any two points in v is not rational. ü) NXE[0,1].0 ] C, EV such that x = C₂ + q where q is a rational no. claim: ucv) is not defined. Proof: W V is mecswable. and A be any bounded, countably infinite set of rational numbers, then from property , {a+v} aes is a disjoint collection of cosets Then we get, M(V)=0 (by (acting using the translation invaiant and countably additive toroperly we get SS ( (a+v)) = { ula+v) = 0 (Since licu) :o) 20

And from property (i), we get- if ve [0, 1] then I cev & [0, 1] with ex=C+9 Since x, ce [0 ] we get q E EL OB. So take, A = [4,4] then, korok Goldtopad [0, ] C U (+ V) - NE[-]ng. this is a contradiction, caurse il [0,1])=1 and we have seena u (ula+v)). = 0. DEA subset a set with positine outer measure Can not be a of a set of measure zero. hence, v is not measurable. i MV) does not exist. (option (as Correct) Scanned with CamScanner