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The Point Of The BanachTarski Theorem  2015/06/06
Added after seeing some discussion ...
This is not intended to explain why the BanachTarski theorem
is true, nor to give a proof, nor to talk about what the pieces
look like, etc. The purpose of this article is to explain why
the result is relevant. In short, it tells us what is and is not
possible in measure theory. It tells us that things we would in
general want to be true of a measure are mutually inconsistent.
Now read on ...

There's a classic "Limited Audience" joke/riddle that goes:
 Q: What's an anagram of BanachTarski?
 A: BanachTarski BanachTarski
Now, if you already know what the BanachTarski theorem says, that
riddle is really funny. If you don't then you're simply not in
the audience, and you'll just go: "Huh?"
Which is a perfectly reasonable reaction. Indeed, if you then have
the BanachTarski theorem explained to you, you most likely will still
go: "Huh?" It's a perplexing result, some even call it a paradox, but
the fact that it's so odd actually masks the fact that it's a truly
important result, with some deep implications.
So I'm going to explain here why it's important, and what some of the
implications are. The hope is that even if you do know the result you
will find this interesting, largely because in the shock of seeing
what the result says, you've never been shown why it's interesting.
So for those of you who don't know the result, here it is in simple,
nontechnical terms:
 In $\Re^3$, given a solid ball $B$ of radius $R,$
it is possible to partition $B$ into finitely many
pieces such that those pieces can be reassembled
to form two solid balls $B_1$ and $B_2,$ each of
radius $R.$
Apart from
using the
Axiom of Choice. 
Now, that's obvious nonsense, and that's why the result is so shocking.
It defies common sense, and immediately makes you go looking for some
kind of loophole. But there isn't one.
So in part because it's so surprising, and shocking, and nonsensical,
you might think it's in a deadend of mathematics and of no real use
or interest. That's what this article is intending to address.
To do so, let's consider the idea of measurement. We can talk about
the length of a line, the area of a polygon (or other shape), the
volume of a lump, or whatever. The development of the concept of
accurate measurement goes back millennia, and is critical in the
development of commerce, engineering, and so many other things. So
we're going to look at the concept of measuring something, and see
what we can say about it mathematically.
To do that we'll talk about a function called a "measure". One of the
problems in maths is we use ordinary words in a technical sense, so
it's a bit dangerous to call this thing a "measure," but I'll try to
use that word only ever in its technical sense.
And fairly obviously we'll need a different function, or measure, when
we're in one dimension as compared with two dimensions, or more, so
we'll talk about a measure for each dimension.
So a "measure" is a function that takes a set and returns a
number that somehow represents the length, area, volume, whatever. 
But regardless of dimension there are things we expect a measure
to do, ways we expect it to behave. For example, we expect the
measure of a zero length line to be zero. We also expect the
measure of a nonzero length line to be nonzero. These seem to
be obvious requirements. Also, when we're talking about length (or
area or volume, etc ) we would expect the sum of the measure of the
parts to be the same as the measure of the whole. In other words, if
you take the measure of a set, then partition the set into two pieces
and take the measure of each of those, you'd expect the sum of the
measures to be the measure of the original.
Let's start writing these down. If we are working in $n$ dimensions
and we have a measure, $\mu$ which is defined on subsets of $\Re^n,$
we expect that:
 $\mu(\{\})\;=\;0$  the measure of the empty set is 0.
 If I is the unit object, then $\mu(I)=1.$
 For sets $A$ and $B$ that do not overlap, then
$\mu(A{\cup}B)\;=\;\mu(A)+\mu(B).$
If we want to, we can weaken the middle condition to simply ask that
$\mu(I)\;>\;0,$ because then we can just apply a scaling. This is like
changing units.
The last one we can repeat over and over to get the idea of the
measure being finitely additive. In fact, we'd really like to
extend that to being countably additive, so that:
 $\mu(\bigcup_{i=0}^{\infty}A_i)\;=\;\sum_{i=0}^\infty\mu(A_i)$
 Provided all the sets are (pairwise) disjoint
In other words, when
we have a set that's made up of countably infinitely many disjoint
pieces, we can choose to take the measure of them all and add them up,
or we can think of the union of them all, and take the measure of that.
But there are other things we expect to be true of a measure. We expect
that if we take the measure of something, move the something around,
and then take the measure again, we get the same answer. Moving
something around should not change its size. In mathematics the idea
of moving something around is captured by the idea of what we call an
isometry. An isometry is a function that doesn't change distances,
so if we have an isometry $\tau$ and apply it to a set $A$, none of the
distances between points in $A$ will change, so we can think of $\tau(A)$
as being $A$ moved somewhere else (and maybe flipped over to give a
mirror image).
So for any set $A,$ and any isometry $\tau,$ then we expect of a
measure $\mu$ that:
 $\mu(A)\;=\;\mu(\tau(A))$
In words, we expect a measure to be isometry invariant.
So far:
 $\mu(I)\;=\;1$
 $\mu$ is countably additive
 $\mu$ is isometry invariant
Finally, we'd like $\mu$ to be defined for all sets, although it
might end up being infinite if our set is unbounded (and even then
not always). So if we have a bounded set (and I've not given a
technical definition of what that means) then we'd like the measure
of that to exist. So we finally want:
 For all bounded sets $A$, $\mu(A)$ exists.
Well, the problem is that we can't have all of these. This was
shown in 1905 and is a classic example. In short, these natural
and obvious requirements for a measure are mutually inconsistent.
We cannot have all of:
 $\mu(I)\;=\;1$
 $\mu$ is countably additive
 $\mu$ is isometry invariant
 For all bounded sets $A$, $\mu(A)$ exists.
Outline of the
Vitali Set proof
The idea is:
 Consider the points from 0 (inclusive) to 1 (not inclusive)
 Declare two points to be equivalent if they differ by a rational
 This gives us families of points, the members of a family
differing only by a rational number.
 Points from different families do not differ by a rational.
 From each equivalence class (family) choose one element
 Call that set $V$.
 Given two points in $V$ they do not differ by a rational
 Enumerate the rationals in $[0,1)$ :
 $r_0,\;r_1,\;r_2,\;\ldots$
 For each rational $r_i$ consider $V_i\;=\;V+r_i\;\;(mod\;1)$
 These are disjoint
 (Exercise: you should check that)
 For any measure $\mu$, for all $i$, $\mu(V_i)\;=\;\mu(V).$
 This requires some proof, but basically $V_i$ is just
$V$ moved about.
 By the properties we have:
 $\mu([0,1))\;=\;\sum_i\mu(V_i)$
But what can $\mu(V)$ possibly be?
If $\mu(V)\;=\;0$ then when we add them all up we get zero, which
would mean the measure of $[0,1)$ is zero.
Which is wrong.


If $\mu(V)\;\ne\;0$ then when we add infinitely many together
we definitely don't get a finite answer.
Which is wrong.

So either way, using only the things we decided were obvious and natural
for a measure, we've arrived at a contradiction. So we cannot have a
measure that has all the properties that we quite reasonably want.

The proof is simple, and can be found all over the 'net. Look
for the
Vitali Set.
So what do we do?
Well, what we have to do is relax one of our requirements, and make it
weaker. The obvious thing that people want to try is to reduce the
power of the additivity rule. So our requirements become:
 $\mu(I)\;=\;1$
 $\mu$ is finitely additive
 $\mu$ is isometry invariant
 For all bounded sets $A$, $\mu(A)$ exists.
Can we do this?
As it happens, for $\Re^1$ we can do this. Even more, for $\Re^2$ we can
do this! But for $\Re^3,$ we can't.
How do we know? Because of the BanachTarski theorem.
The BanachTarski theorem says that if $B$ is the unit ball in $\Re^3$,
there exist pairwise disjoint sets $\{A_i\}_{i=1}^n$ and isometries
$\{\tau_i\}_{i=1}^n$ and $\tau$ such that:
 $\bigcup_{i=1}^nA_i\;=\;B$
 $B\cap\tau(B)=\{\}$
 $\bigcup_{i=1}^n\tau_i(A_i)\;=\;B\cup\tau(B)$
In other words, $B$ can be partitioned into sets, and those sets can be
moved about and reassembled to make two balls the same as the original.

That means there can be no measure satisfying the
requirements even when weakened from countable
additivity to finite additivity.



It's
not enough in three dimensions.
And that's what the BanachTarski theorem is really all about.
There's more we can say about this. Do we want or need every set to
have a measure? We can define some really weird sets  should it always
make sense for them to have a concept of length/area/volume?
There is one thing that could save us, and that's the Axiom of Choice.
Or rather, denying the Axiom of Choice. In each case, showing that in
$\Re^1$ there's an unmeasurable set, and showing that in $\Re^3$ we can
have a paradoxical decomposition, we need to use the Axiom of Choice.
So maybe we should choose not to believe in the Axiom of Choice.
But that's another discussion.
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