On Mon, Jun 6, 2011 at 1:11 PM, Dan Asimov <dasimov@earthlink.net> wrote:
About 10 years ago I devised a construction, based on the Axiom of Choice, that results in the selection of a random integer under the assumption that it makes sense to pick a random point from the circle R/Z -- or equally, the interval [0,1).
Of course Bill is referring to the well-known finitely-additive measure on the integers known as density, which of course is defined only on certain subsets.
And speaking of density, it seems that using special averaging methods like Cesaro summation, one can extend the collection of subsets of Z on which density is defined. My question is,\:
Is there a well-defined maximum collection of such subsets? Or at least maximal ones?
I believe that what Bill is saying is that density can be extended to a finitely additive translation-invariant measure defined on all sets of integers. But there's no canonical way to do this, and no such way can be explicitly specified, and proving the existence of this extension requires use of AC. Andy
--Dan
Bill T. 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.
Sometimes the brain has a mind of its own.
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