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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?

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 $\Sigma_\Pi$.

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# 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?