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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.

It reads:

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
up vote 1 down vote accepted

This reference seems to be exhaustive (and exhausting) on the subject:

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Thank you Igor, that is exactly what I was after. – Daniel Mansfield Aug 26 '11 at 23:33

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