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