Quotients of Measurable Spaces? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T23:01:59Z http://mathoverflow.net/feeds/question/71407 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71407/quotients-of-measurable-spaces Quotients of Measurable Spaces? Michael Greinecker 2011-07-27T14:32:20Z 2012-02-08T19:06:37Z <p>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 $\Sigma_\Pi=\{A\subseteq\Pi:\cup A\in\Sigma\}$.</p> <p>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:</p> <blockquote> <p>Are there any papers or texts that study this quotient construction and its properties? Are there other commonly used quotient constructions for measurable spaces?</p> </blockquote> <p>Edit: Additional question:</p> <blockquote> <p>What are sufficient conditons for $\Sigma_\Pi$ to be countably generated?</p> </blockquote> <p>The problem here is that generators for $\Sigma$ cannot simply be transferred to generators of $\Sigma_\Pi$.</p>