From: rbisrael@my-deja.com Subject: Re: Looking for an A in [0,1], that is not in the Borel-Sigma-field Date: Tue, 25 Apr 2000 00:26:00 GMT Newsgroups: sci.math Summary: [missing] In article , "M. Vaeth" wrote: > Let V be a nonmeasurable set obtained by the classical Vitaly construction > (i.e. you call two numbers of [0,1] equivalent if their difference is a > rational number; let V be a set which contains from each equivalence class > precisely one element (AC!)). > The set V has inner measure 0 and outer measure m>0. There is some natural > number n with m>1/n. Now, choose n different rationals q_1,...,q_n in [0,1), > let V_j=V+q_j (calculate modulo 1), and let A be the union of the disjoint > sets V_j. > However, in the moment I can neither prove that A has outer measure 1 nor > that it has inner measure 0; maybe the above construction has to be refined > (e.g. by a proper choice of q_j, etc). Let G be a dense subgroup of the rationals Q such that Q/G is infinite, and let U = V + G (mod 1). Then U has inner measure 0 and outer measure 1. Robert Israel israel@math.ubc.ca Sent via Deja.com http://www.deja.com/ Before you buy.