Re: [math-fun] Paradox
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? --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.
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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-- Andy.Latto@pobox.com
On Jun 6, 2011, at 1:35 PM, Andy Latto wrote:
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.
That is correct. In contrast, it is impossible to define a finitely additive measure on the free group on two generators that is invariant under left multiplication by an element of the group. This is related to the Banach-Tarski paradox. A group admits such a measure if and only if it is amenable. A more geometric and more constructive definition of amenability is this: a finitely generated group is amenable if and only if for every epsilon, there is a finite subset S of the group such that for each generator g, the symmetric difference of S and gS has cardinality less than epsilon times the cardinality of S. For many amenable groups (including Z or Z^n) but not all amenable groups, you can take S to be the set of all elements of the group expressible with word length less than R, for some large R. There are other nice characterizations of amenability as well, there is a large literature on amenability, and there are a number of interesting and challenging open problems. Amenability is an interesting topic, but it's wandering far from the original question and not necessarily in the spirit of math-fun. Bill
participants (3)
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Andy Latto -
Bill Thurston -
Dan Asimov