Date: Feb 16, 2013 10:56 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.
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, for
any 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.