# Point mapping induces a set mapping

Mathematics is the universal language.

That is, until someone says the word "obvious", or "well known". At which point it becomes relative to the reader.

My question is about a "well known" theorem. My problem is that it is not known to me. But I would like to know.

The following comes from Y. Katznelson and B. Weiss The Classification of Non-Singular Actions, Revisited, J. Ergodic Theory and Dynamical Systems, 11, 1991. Page 4.

Thus the family {$\theta_k$} defines a Boolean set mapping between the $\sigma$-algebras generated by the ladder sets. If the ladder sets are both "algebra complete", a well known theorem implies that there exists a point mapping $\theta : X \mapsto X^\prime$ which induces a set mapping.

Can someone please tell me which theorem they are referring to here?

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I'm not sure what theorem, but quickly scanning the paper, the condition 'both ladder sets are algebra complete' seems to imply that the family ${\theta_k\}$ defines a mapping of Boolean algebras $B \to B'$. The theorem you are looking for is probably something along the lines of: given a mapping of Boolean algebras $B\to B'$ where $B$ and $B'$ are $\sigma$-algebras of finite measure spaces $X$ and $X'$ resp., one has a map $X\to X'$ of the underlying sets (finite because this is what the authors assume). Finiteness here makes this much easier. But I'm not an expert in this field. – David Roberts Aug 26 '11 at 2:07
Maybe I'm mistaken about the finiteness making things easier. :) – David Roberts Aug 26 '11 at 2:09