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The Doob-Dynkin Lemma gives conditions on two functions $X$ and $Y$ thatensure existence of a function ${\phi}$ so that $X = {\phi} \circ Y$. Thiscommunication proves different versions of the Doob-Dynkin Lemma, and shows howit is related to optimal statistical learning algorithms.  Keywords and phrases: Improper prior, Descriptive set theory, ConditionalMonte Carlo, Fiducial, Machine learning, Complex data.

Optimal Learning from the Doob-Dynkin lemma