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, 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.

quasi