Rotwang
Posts:
1,678
From:
Swansea
Registered:
7/26/06


Re: Measure and Density
Posted:
Feb 16, 2013 10:16 PM


On 17/02/2013 02:33, William Elliot wrote: > Topology Q+A Board Ask An Analyst > > 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? > > I have thought of fat Cantor sets, but I cannot see well how to > do it. Any suggestions, please?
Let (q_n  n in N) be an enumeration of the rationals in [0, 1], and consider the union of the sets (q_n  1/2^{n + 3}, q_n + 1/2^{n + 3}) n [0, 1] for n in N. I think that works.
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