tag:blogger.com,1999:blog-3891434218564545511.post804034158136073221..comments2024-03-27T20:37:09.185-05:00Comments on Alexander Pruss's Blog: Uniform measure and nonmeasurable sets, without the Axiom of ChoiceAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-3891434218564545511.post-78368272320043141882013-10-25T06:39:05.871-05:002013-10-25T06:39:05.871-05:00I meant: on all subsets of [0,1).I meant: on all subsets of [0,1).Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-83155425393428799252013-10-25T06:11:12.085-05:002013-10-25T06:11:12.085-05:00This comment has been removed by the author.Matthttps://www.blogger.com/profile/09693352428090978035noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-49259495777565499802013-10-25T06:10:58.436-05:002013-10-25T06:10:58.436-05:00Could you please explain the first sentence? Sure...Could you please explain the first sentence? Surely Lebesgue measure is translation-invariant in the requisite sense, no?Matthttps://www.blogger.com/profile/09693352428090978035noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-43014805793909822942012-12-11T10:13:36.340-06:002012-12-11T10:13:36.340-06:00Slight generalization.
Let P be a finitely additi...Slight generalization.<br /><br />Let P be a finitely additive measure on the countable subsets of some set Omega taking values in some partially ordered group G. Let H be a group that acts on Omega such that there are g in H and w in Omega with { g^n w : n in N } infinite.<br /><br />Then (a) P assigns null weight to some non-empty sets (i.e., P is not regular) or (b) P is not H-invariant.<br /><br />If H acts transitively on Omega, then (a) implies P assigns null weight to all singletons.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.com