
Re: Measure and Density
Posted:
Feb 19, 2013 3:37 PM


On Feb 19, 1:39 pm, W^3 <82nd...@comcast.net> wrote: > In article <Pine.NEB.4.64.1302161828050.5...@panix2.panix.com>, > 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? > > > I have thought of fat Cantor sets, but I cannot see well how to > > do it. Any suggestions, please? > > More interesting is to require 0 < m(S /\ I) < m(I) for all nonempty > open intervals I contained in (0,1).
Let I_0, I_1, I_2, . . . be an enumeration of the open intervals with rational endpoints. Construct a sequence of pairwise disjoint sets F_0, F_1, F_2, . . . so that F_n is a nowhere dense set of positive measure contained in the interval I_{floor(n/2)}. Let S be the union of the sets F_n where n is even.

