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. I must be misinterpreting some part of your terminology, but I'm not sure what. Andy