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Let $(\Omega,\Sigma)$ be a measurable space and $\Pi$ be a partition of $\Omega$. There is a projection $\pi:\Omega\to\Pi$ that maps each $\omega\in\Omega$ to the unique partition cell in $\Pi$ containing $\omega$. We can endow $\Pi$ with the largest $\sigma$-algebra $\Sigma_\Pi$ that makes $\pi$ measurable. It is easily shown that $\Sigma_\Pi=\{A\subseteq\Pi:\cup A\in\Sigma\}$.

This seems to be the most natural way to construct a quotient of a measurable space. I'm sure someone must have used this construction before, but I couldn't find a single paper making use of it. In general, outside of statistical decision theory and topological measure theory, there seems to be little work on measurable spaces in themselves. To focus:

Are there any papers or texts that study this quotient construction and its properties? Are there other commonly used quotient constructions for measurable spaces?

Edit: Additional question:

What are sufficient conditons for $\Sigma_\Pi$ to be countably generated?

The problem here is that generators for $\Sigma$ cannot simply be transferred to generators of $\Sigma_\Pi$.

Michael Greinecker
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    Rokhlin's book "On the fundamental ideas of measure theory" develops a theory of measurable partitions... – Anthony Quas Jul 27 '11 at 17:18
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    Sometimes they just look at one of these as a counterexample. The quotient sigma-algebra for $\mathbb R / \mathbb Q$ . Or the "tail" sigma-algebra in a product $\prod_{n=1}^\infty T_n$ where the factors $T_n$ are nice. Or the "countable subsets of $\mathbb R$ ", realized as the sequences $\mathbb R^{\mathbb N}$ modulo the permutations. A point is: if it is not countably separated, then such a sigma-algebra is very bad in a sense that logicians will tell you about. – Gerald Edgar Jul 27 '11 at 17:45
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    Rohlin's book is very intersting. It can be found at: http://ma.huji.ac.il/~matang02/rohlin.pdf – Michael Greinecker Feb 08 '12 at 19:12
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    In the special case that $\Sigma$ is standard Borel (generated by a Polish topology), this sort of question has been extensively studied among descriptive set theorists. For example, if $\Pi$ is a partition arising from a Borel equivalence relation, the quotient $\sigma$-algebra is countably generated if and only if there is a Borel assignment of real invariants to the equivalence classes. In the special case that the equivalence relation has countable classes (among other such special cases), this is equivalent to finding a Borel set which intersects each class in exactly one point. Etc. – Clinton Conley Feb 08 '12 at 19:36
  • Note that if your original $\Sigma$ is countably generated and separates points, it is automatically standard Borel, so the previous comment applies. – Clinton Conley Feb 08 '12 at 19:44
  • Thank you! Could you recommend a reference (as a starting point) for these kinds of results? – Michael Greinecker Feb 08 '12 at 19:45
  • A countable generated $\sigma$-algebra that separates points is not necessarily standard Borel. If the continuum hypothesis is wrong, we can take the trace $\sigma$-algebra of the Borel $\sigma$-algebra of an uncountable set of real numbers with cardinality less than $\mathfrak{c}$. – Michael Greinecker Feb 08 '12 at 19:56
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    A good default reference for properties of standard Borel spaces would be Kechris' descriptive set theory text. More recent references about equivalence relations in particular include Kanovei's Borel Equivalence Relations and Gao's Invariant Descriptive Set Theory. Also noteworthy is the Jackson, Kechris, Louveau paper entitled "Countable Borel equivalence relations," J. Math. Logic. – Clinton Conley Feb 08 '12 at 19:59
  • You are right; I meant to say "restriction of a standard Borel $\sigma$-algebra." I'm not having good luck posting coherently lately. – Clinton Conley Feb 08 '12 at 20:03
  • I think this question of mine is related, though I am not sure whether the maximal/minimal $\sigma$-algebra would work there. – SBF Jul 18 '14 at 11:48
  • @MichaelGreinecker The link for the Rohlin book isn't working. What was the book? – Aidan Young May 19 '22 at 18:09
  • @AidanYoung "On the fundamental ideas of measure theory". It's more of a booklet though; 54 pages in total. – Michael Greinecker May 19 '22 at 18:32

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