Speaking of craziness, what about measures with nonstandard (in the sense of hyperreal or surreal) values? It seems natural to assign a uniform probability measure where the value of any single integer is some infinitesimal epsilon. The lack of the Archimedean property means that the measure of {1, 2, 3, ... n} can still be less than one for any n. In particular, I wonder if you can get one of the translation-invariant but non-countably-additive measures that Bill is talking about by taking the standard part of such a nonstandard uniform measure on Z. -Thomas C On Mon, Jun 6, 2011 at 6:50 AM, Bill Thurston <wpthurston@mac.com> wrote:
On Jun 6, 2011, at 1:41 AM, Andy Latto wrote:
On Sun, Jun 5, 2011 at 9:32 PM, Bill Thurston <wpthurston@mac.com> wrote:
====== Non countably-additive measures are legitimate structures, and provide one of the ways to define amenability for a group, but they are totally weird and I don't think you really want to go into that territory. In theory, there are translation-invariant additive but non-countaby-additive measures on Z, but it is known to be impossible to actually define any particular instance of one.
How can this be?
Let the measure of {1} be x > 0, and choose an integer p such that x > 1/p.
If the measure is translation invariant, then the measure of any singleton is x.
But if the measure is finitely additive, then the measure of {1,2,3,...p} is xp > 1, so the measure can't be a probability measure.
You've given a correct deduction that the measure of any singleton, and in fact the measure of every finite set is 0. The measure must be 1 for every set whose complement is finite. The measure must be 1/2 for all even integers, and 1/2 for all odd integers, etc --- there are certain similar things you can deduce. The craziness comes in extending these definitions to *all* subsets of the integers.
Bill
I must be misinterpreting some part of your terminology, but I'm not sure what.
Andy
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun