```Date: Feb 16, 2013 10:56 PM
Author: quasi
Subject: Re: Measure and Density

quasi wrote:>William Elliot wrote:>>>> [User "Herb" on forum "Ask An Analyst" asked]:>>>>How can we find a measurable dense subset S of [0,1], with >>m(S) < 1, and such that for any (a,b) in [0,1], we have>>m(S /\ (a,b)) > 0?>>Let Q denote the set of rational numbers and let>>   x_1, x_2, x_3, ... >>be an enumeration of Q /\ (0,1). >>For each positive integer k, let>>  a_k = max(0,x_k - 1/(2^(k+1)))>>  b_k = min(1,x_k + 1/(2^(k+1)))>>and define the open interval I_k by>>  I_k = (a_k,b_k)>>Finally, let S be the union of the intervals >>  I_1, I_2, I_3, ...>>Then S satisfies the required conditions.Now that I see Rotwang's solution (our solutions are essentially the same with a minor difference), I realize that I should have used  a_k = max(0,x_k - 1/(2^(k+2)))  b_k = min(1,x_k + 1/(2^(k+2)))to make it more evident that m(S) < 1.However, my original solution was not actually wrong since, forany enumeration of Q /\ (0,1), the intervals    I_1, I_2, I_3, ...cannot be pairwise disjoint, hence it's still true that m(S) < 1.quasi
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