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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 Schröder–Bernstein theorem for measurable spaces ： ウィキペディア英語版 Schröder–Bernstein theorem for measurable spaces The Cantor–Bernstein–Schroeder theorem of set theory has a counterpart for measurable spaces, sometimes called the Borel Schroeder–Bernstein theorem, since measurable spaces are also called Borel spaces. This theorem, whose proof is quite easy, is instrumental when proving that two measurable spaces are isomorphic. The general theory of standard Borel spaces contains very strong results about isomorphic measurable spaces, see Kuratowski's theorem. However, (a) the latter theorem is very difficult to prove, (b) the former theorem is satisfactory in many important cases (see Examples), and (c) the former theorem is used in the proof of the latter theorem. ==The theorem== Let $X\backslash ,$ and $Y\backslash ,$ be measurable spaces. If there exist injective, bimeasurable maps $f\; :\; X\; \backslash to\; Y,\; \backslash ,$ $g\; :\; Y\; \backslash to\; X,\; \backslash ,$ then $X\backslash ,$ and $Y\backslash ,$ are isomorphic (the Schröder–Bernstein property). 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「Schröder–Bernstein theorem for measurable spaces」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.