# Quotients of Measurable Spaces?

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?

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 $\Sigma_\Pi$.
• 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. Jul 27, 2011 at 17:45
• 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. Feb 8, 2012 at 19:36