most recent 30 from http://mathoverflow.net 2013-06-19T17:53:16Z http://mathoverflow.net/feeds/question/109990 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109990/are-all-compact-groups-amenable Feldmann Denis 2012-10-18T09:16:38Z 2012-10-18T09:37:21Z

<p>Wikipedia states that the Haar measure on a compact group is a mean (and that every compact group is amenable). But, obviously, the Haar mesure on the group of unit quaternions cannot be defined on every subset, else the Banach-Tarski paradox would not happen. What am I missing?</p>

http://mathoverflow.net/questions/109990/are-all-compact-groups-amenable/109997#109997 HW 2012-10-18T09:37:21Z 2012-10-18T09:37:21Z

<p>The definition of a mean for a locally compact group is <em>not</em> a finitely additive measure defined on the entire power set of $G$---that definition is only correct for discrete $G$. See <a href="http://en.wikipedia.org/wiki/Amenable#Definition_for_locally_compact_groups" rel="nofollow">the wikipedia article</a> for the correct definition. Indeed, the definition given there refers to the Haar measure, so is easily seen to hold in the compact case.</p> <p>For indiscrete $G$ there will typically be non-measurable subsets; of course, the pieces that appear in the Banach--Tarski paradox are non-measurable.</p>