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Topic: Measure and Density
Replies: 14   Last Post: Feb 23, 2013 11:26 AM

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Posts: 12,067
Registered: 7/15/05
Re: Measure and Density
Posted: Feb 16, 2013 10:31 PM
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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.


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