From: israel@math.ubc.ca (Robert Israel)
Subject: Re: An Obtuse Paradox?
Date: 4 Sep 2000 23:09:29 GMT
Newsgroups: rec.puzzles,sci.math

In article <8p0rhg$vn1$1@news.fas.harvard.edu>,
Noam D. Elkies  <elkies@maf'th.harvard.edu> wrote:

>ObPuzzle: Suppose F is a family of negligible subsets of [0,1]
>such that if S and S' are in F then either S contains S' or
>S' contains S.  Must the union of all the sets in F be negligible?

(where "negligible" = "measure 0")

No, assuming the Axiom of Choice.
Let A be the least ordinal such that there exists a subset
of [0,1] of cardinality |A| that is not negligible, and let E be
a non-negligible subset of cardinality |A|.  Let <= be an ordering
of E corresponding to the ordinal A.  The family 
F consists of the sets S_x = { y in E: y <= x } (in terms of the 
well-ordering) for x in E.  Then each S_x has cardinality < |A| 
and thus is negligible, but their union is E which isn't.
And for any two sets S_x and S_y, one contains the other (depending
on how x and y compare in the well-ordering).

 
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2
==============================================================================

From: Noam D. Elkies  <elkies@aleph_null.harvard.edu> wrote:
Subject: Re: An Obtuse Paradox?
Date: 5 Sep 2000 15:49:09 GMT
Newsgroups: rec.puzzles,sci.math
Summary: Another ObPuzzle spoiler

In article <8p1a39$ll3$1@nntp.itservices.ubc.ca>,
Robert Israel <israel@math.ubc.ca> wrote:
>In article <8p0rhg$vn1$1@news.fas.harvard.edu>, [I wrote:]

>>ObPuzzle: Suppose F is a family of negligible subsets of [0,1]
>>such that if S and S' are in F then either S contains S' or
>>S' contains S.  Must the union of all the sets in F be negligible?

>(where "negligible" = "measure 0")

>No, assuming the Axiom of Choice.

Right.

>Let A be the least ordinal such that there exists a subset
>of [0,1] of cardinality |A| that is not negligible, and let E be
>a non-negligible subset of cardinality |A|.  [...]

An even simpler argument:

If the answer was Yes then the family of *all* negligible subsets
of [0,1] would satisfy the hypotheses of Zorn's Lemma!  There would
thus be a maximal negligible subset S; since the union of S with {x}
is then still negligible for any x in [0,1], it follows that S is
all of [0,1] -- which, however, is not negligible, QED.