定義

設是一(完備)機率空間。

定義1 如果對於任何,有



則空間到自身的影射稱做 可測的。

定義2 如果對於任何,有


則可測影射稱做 保測變換(morphism)。




設T是保測變換,是其n次冪,而是一隨機變數,設,考慮序列。
保測變換舉例








例1 設是由有限個點構成的集合,是中一切子集的代數,,而.如果,則是保測變換。



例2 設, P是勒貝格測度,,則是保測變換。
保測變換的物理前提條件












假設某一系統按照一定運動規律(在離散時間)演變,並構想是該系統的狀態的相空間,那么,如果是系統在時刻的狀態,則是經過n步系統進入的狀態,其中是(該運動規律誘導的)推移運算元,其次,假如A是某一“狀態的集合”,則根據其定義是經一步到達集合A的一切“初始”狀態的集合,因此,假如把視為“不可壓縮的液體”,則條件可以視為完全自然的“體積”保持不變的條件。(對於經典的封閉哈密頓(W.R.Hamilton)系統,著名的劉維爾(J.Liouville)定理斷定,相應的變換是保持勒貝格測度不變的變換。)
關於保測變換的龐加萊定理
下面關於“常返性”的龐加萊(J.H.Poincaré)定理(1912),是有關保測變換最早的成果之一。





定理 設是一機率空間, T是保測變換,.那么,對於無限多個和幾乎一切點,有。







證明: 記={,對於一切).由於對於任意,,則,這樣,序列由有相同 P一測度的不相交集合構成,因此,





從而。從而,對於幾乎一切點和至少對於一個,由此可見,對於無限個n≥1,有。






將上面得到的結果用於變換,那么,對於每一個點\,其中 N是0機率集合(並且 N是對應於不同 k的相應集合的並),存在這樣的,使.由此顯然可以得到,對於無限多個n,有。


設,則在集合上


如果令當時,得所要證明的結果。

注: 假如將機率測度 P換成任意有限測度,則定理仍然成立。