From: Fred Galvin
Subject: Re: Banach-Tarski in <3 dimensions
Date: Sat, 10 Apr 1999 02:55:43 -0500
Newsgroups: sci.math
On Sat, 10 Apr 1999, Jonathan W. Hoyle wrote:
> Yes, Fred, you constructed a non-measurable set correctly, but you have
> not shown how it can be recombined to generate a set of measure two.
I thought I had shown that. I'm sorry to hear that my attempt was
unsatisfactory; let me try again.
1. Let me introduce some ad hoc and rather inappropriate terminology,
because I'm too tired to think of anything better. If A and B are sets of
real numbers and n is a cardinal number, I will call a function f:A-->B an
"n-map" if f is a bijection and the set {f(x)-x: x in A} has cardinality
less than or equal to n. Also, I will say that A and B are "n-equivalent"
if there is an n-map f:A-->B. (I realize that "n-equivalence" is not a
transitive relation; however, it is symmetric.)
2. I understand your problem "decompose the unit line segment [0,1] into a
countable number of subsets which can be recombined into a set of measure
2" to mean "find a set B of measure 2 and an omega-map f:[0,1]-->B".
3. I propose to accomplish your task using B = [0,2]. It will suffice to
construct an omega-map f:[0,1)-->[0,2).
4. Use the axiom of choice to get a subset S of [0,1) with the property
that, for each y in [0,1), there is one and only one x in S such that y-x
is rational.
5. Let T be the set of all rational numbers in [0,1). For each t in T, let
S(t) = S+t mod 1 = {x in [0,1): x-t is in S, or x-t+1 is in S}.
6. The sets S(t), t in T, are pairwise disjoint sets whose union is [0,1).
7. Moreover, the sets S(t) are pairwise 2-equivalent. Namely, if t,u in T,
and t < u, we define a 2-map f:S(t)-->S(u) by setting f(x) = x+u-t if
x+u-t < 1 and f(x) = x+u-t-1 otherwise.
8. Since the set T is countably infinite, we can reindex the collection of
S(t)'s (without repetition) as A(1), A(2), ..., A(n), ... . Thus, the sets
A(n), n = 1,2,3,... are pairwise disjoint and pairwise 2-equivelent sets
whose union is [0,1).
9. For each odd index n, define a 2-map f_n:A(n)-->A((n-1)/2).
10. For each even index n, define a 2-map f_n:A(n)-->A(n/2)+1.
11. The set-theoretic union of the 2-maps f_n, n = 1,2,3,... is an
omega-map f:[0,1)-->[0,2), which we can extend to an omega-map
f:[0,1)-->[0,2) by defining f(1) = 2.
12. In a similar way, by partitioning the index set into countably many
subsets instead of only two, we can get an omega-map of [0,1) onto the
whole real line, as I mentioned in my previous message.