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In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin, characterizes the situation when one random variable is a function of another by the inclusion of the -algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the -algebra generated by the other. The lemma plays an important role in the conditional expectation in probability theory, where it allows to replace the conditioning on a random variable by conditioning on the -algebra that is generated by the random variable. ==Statement of the lemma== Let be a sample space. For a function , the -algebra generated by is defined as the family of sets , where are all Borel sets. Lemma Let be random elements and be the algebra generated by . Then is -measurable if and only if for some Borel measurable function . The "if" part of the lemma is simply the statement that the composition of two measurable functions is measurable. The "only if" part is the nontrivial one. By definition, being -measurable is the same as for any Borel set , which is the same as . So, the lemma can be rewritten in the following, equivalent form. Lemma Let be random elements and and the algebras generated by and , respectively. Then for some Borel measurable function if and only if . 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「Doob–Dynkin lemma」の詳細全文を読む スポンサード リンク
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