[FOM] Query concerning measure.

[Harvey Friedman]

On 2/18/06 10:47 PM, "Bill Taylor" <W.Taylor at math.canterbury.ac.nz> wrote:

> Query related to the current threads:
> 
> Is it consistent with ZFC that there is a finitely additive set function
> on ALL subsets of  [0,1)  that agrees with length for intervals,
> and is translation (i.e. rotation) invariant?
> 
Yes. In modern terminology, one shifts from

1) there exists a finitely additive nonnegative set function on all subsets
of [0,1) that is translation invariant and measures [0,1) as 1

to

2) the group G of translations of [0,1) is amenable.

I.e., 

3) there exists a G invariant finitely additive nonnegative set function on
all subsets of G that measures G as 1.

There is a standard result that every Abelian group is amenable. See, e.g.,
http://en.wikipedia.org/wiki/Amenable_group
Amenability of Discrete Groups

Obviously any measure in 1) must measure the intervals correctly.

Your question also has an affirmative answer in 2 dimensions, but not in 3
dimensions. See 

S. Wagon, The Banach Tarski Paradox, Encyclopedia of Mathematics and its
Applications, Cambridge University Press, 1985.

By the way, I found this quote (not clear just from where) at the bottom of
the webpage 

http://www.cs.uwaterloo.ca/~alopez-o/math-faq/node70.html

stating the following:

"Banach and Tarski had hoped that the physical absurdity of this
theorem would encourage mathematicians to discard AC. They were
dismayed when the response of the math community was 'Isn't AC great?
How else could we get such counterintuitive results?' "

Comments please on this quote.

Harvey Friedman