Date: Feb 16, 2013 10:31 PM
Subject: Re: Measure and Density
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.