From: Fred Galvin <galvin@math.ukans.edu>
Subject: Re: MEASURE ZERO
Date: Wed, 3 Mar 1999 00:16:44 -0600
Newsgroups: sci.math
Keywords: Undecidability of existence of measure zero sets

On 3 Mar 1999, Alexander Abian wrote:

> Prove or disprove that every uncountable set of real numbers 
> has an uncountable subset of Lebesgue measure zero

Undecidable. On the one hand, assuming ZFC+CH, it's easy to "construct"
counterexamples by transfinite induction: there are only aleph_one G_delta
sets of measure zero, so you can construct a set of aleph_one real numbers
whose intersection with each of those is countable. (The counterexamples
are called Sierpinski sets after their discoverer; they are dual to Lusin
sets under measure-category duality.) On the other hand, there are models
of ZFC in which every set of cardinality aleph_one has Lebesgue measure
zero; of course, by the axiom of choice, every uncountable set of real
numbers has a subset of cardinality aleph_one.