
Re: Measure and Density
Posted:
Feb 16, 2013 11:45 PM


On Feb 16, 8:33 pm, William Elliot <ma...@panix.com> 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?
Any dense open set will have the property that its intersection with each interval has positive measure. Therefore, it will suffice to construct a dense open subset S of (0,1) with m(S) < 1. In fact, you can easily make your dense open set S have measure less that an arbitrary positive number epsilon. If you do that for an infinite sequence of epsilons tending to zero, and take the intersection, you get a dense G_{delta} set of measure zero. Haven't you seen that construction?

