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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Følner sequence ： ウィキペディア英語版 Følner sequence In mathematics, a Følner sequence for a group is a sequence of sets satisfying a particular condition. If a group has a Følner sequence with respect to its action on itself, the group is amenable. A more general notion of Følner nets can be defined analogously, and is suited for the study of uncountable groups. Følner sequences are named for Erling Følner. == Definition == Given a group $G$ that acts on a countable set $X$, a Følner sequence for the action is a sequence of finite subsets $F\_1,\; F\_2,\; \backslash dots$ of $X$ which exhaust $X$ and which "don't move too much" when acted on by any group element. Precisely, :For every $x\backslash in\; X$, there exists some $i$ such that $x\; \backslash in\; F\_j$ for all $j\; >\; i$, and :$\backslash lim\_\backslash frac\; =\; 0$ for all group elements $g$ in $G$. Explanation of the notation used above: *$gF\_i\backslash$ is the result of the set $F\_i\backslash$ being acted on the left by $g$. It consists of elements of the form $gf$ for all $f$ in $F\_i$. *$\backslash triangle$ is the symmetric difference operator, i.e., $A\backslash triangle\; B$ is the set of elements in exactly one of the sets $A$ and $B$. *$|A|$ is the cardinality of a set $A$. Thus, what this definition says is that for any group element $g$, the proportion of elements of $F\_i\backslash$ that are moved away by $g$ goes to 0 as $i$ gets large. In the setting of a locally compact group acting on a measure space $(X,\backslash mu)$ there is a more general definition. Instead of being finite, the sets are required to have finite, non-zero measure, and so the Følner requirement will be that * $\backslash lim\_\backslash frac\; =\; 0$, analogously to the discrete case. The standard case is that of the group acting on itself by left translation, in which case the measure in question is normally assumed to be the Haar measure. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Følner sequence」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.