[FOM] outer measure of a non-measurable set

Jeff Hirst jlh at math.appstate.edu
Thu Mar 28 09:46:47 EDT 2013

Hi Martin-

On Mar 28, 2013, at 2:20 AM, Martin Davis <martin at eipye.com> wrote:
A friend asked me. Can anything be said about the outer measure of a set containing one point from each equivalence class with respect to the relation x~y <=> x-y is rational, x,y in [0,1]?

The outer measure of such a "Vitali set" can be anything in (0,1].  Here's a link to a short note that Jonathan Loss and I wrote up:   http://mathsci2.appstate.edu/~jlh/pdf/HL092007.pdf

A very well read referee told us that the result was known.  For example, it is a lemma in the paper "On compactness and Loeb measures" by Alzaz in the Proceedings of the AMS, vol 123, 1995.

Hope this helps…


Jeff Hirst,  Professor of Mathematics
Department of Mathematical Sciences
Appalachian State University
Boone, NC 28608
828-262-2861  jlh at math.appstate.edu

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