10
$\begingroup$

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 $\Sigma_\Pi$.

Improve this question
$\endgroup$
12
  • 4
    $\begingroup$ Rokhlin's book "On the fundamental ideas of measure theory" develops a theory of measurable partitions... $\endgroup$
    – Anthony Quas
    Commented Jul 27, 2011 at 17:18
  • 5
    $\begingroup$ 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. $\endgroup$
    – Gerald Edgar
    Commented Jul 27, 2011 at 17:45
  • 2
    $\begingroup$ Rohlin's book is very intersting. It can be found at: ma.huji.ac.il/~matang02/rohlin.pdf $\endgroup$
    – Michael Greinecker
    Commented Feb 8, 2012 at 19:12
  • 2
    $\begingroup$ 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. $\endgroup$
    – Clinton Conley
    Commented Feb 8, 2012 at 19:36
  • 2
    $\begingroup$ 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. $\endgroup$
    – Clinton Conley
    Commented Feb 8, 2012 at 19:59

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .