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Stream: event: Categorical Probability and Statistics 2020 workshop

Topic: The poset of measures wrt absolute continuity

Tobias Fritz (Jun 08 2020 at 22:22):

Consider the set of finite measures on $[0,1]$ as a preordered set with respect to absolute continuity: $\mu \le \nu$ if and only if $\nu(A) = 0$ implies $\mu(A) = 0$ for every Borel set $A$. What is known about the order-theoretic structure of this preordered set?

It is quite clear that the zero measure is a bottom element and that there is no top element. Using addition of measures, it is also easy to see that binary joins exist. Does bounded directed completeness hold as well? Do binary meets exist?

If binary meets exist, how do they interact with joins? Do we get a distributive lattice?

Paolo Perrone (Jun 08 2020 at 22:42):

I believe that, with this preorder on the spaces of measures, the category of measure spaces and lax-measure-preserving maps forms a genuine Grothendieck opfibration (not just a discrete one) over the category of measurable spaces.

Paolo Perrone (Jun 08 2020 at 22:55):

Tobias Fritz said:

Consider the set of finite measures on $[0,1]$ as a preordered set with respect to absolute continuity: $\mu \le \nu$ if and only if $\nu(A) = 0$ implies $\mu(A) = 0$ for every Borel set $A$. What is known about the order-theoretic structure of this preordered set?

It is quite clear that the zero measure is a bottom element and that there is no top element. Using addition of measures, it is also easy to see that binary joins exist. Does bounded directed completeness hold as well? Do binary meets exist?

If binary meets exist, how do they interact with joins? Do we get a distributive lattice?

Let $\mu$ and $\nu$ be absolutely continuous with respect to $\rho$ (which exists, since there are joins), and denote by $f$ and $g$ their Radon-Nikodym derivatives, respectively. Does the measure $f\,g\,\rho$ give the meet?

Evan Patterson (Jun 08 2020 at 23:55):

In the case of the stronger, set-wise order on finite (signed) measures (given by $\mu \leq \nu$ iff $\mu(A) \leq \nu(A)$ for all $A$), I once checked that meets and joins exist and are given by meets and joins of the corresponding Radon-Nikodym derivatives. Thus, the meet of $\mu$ and $\nu$ is $d(\mu \wedge \nu) = (f \wedge g) d\rho$, where $f$ and $g$ are the derivatives of $\mu$ and $\nu$ with respect to any dominating (nonnegative) measure $\rho$.

Evan Patterson (Jun 08 2020 at 23:59):

I could post some hand-written but readable notes on this, if people are interested. I'm not sure about the case of the absolute continuity order.

Paolo Perrone (Jun 09 2020 at 01:21):

Evan Patterson said:

I could post some hand-written but readable notes on this, if people are interested. I'm not sure about the case of the absolute continuity order.

That would be nice!

Evan Patterson (Jun 09 2020 at 01:46):

OK, here are some notes, which I hope are legible (and correct!). lattice_measures.pdf

Paolo Perrone (Jun 09 2020 at 13:22):

Thank you!

Tobias Fritz (Jun 09 2020 at 14:03):

Thanks, Evan! That construction seems likely to work, in the sense that your meets and joins are plausibly also meets and joins with respect to the absolute continuity preorder. But I haven't gone through the details yet.

Some background for the question: I've been thinking a bit about categories similar to measurable spaces, and I've been wondering whether there may be such a category in which every probability measure on every space would have a "support", either in the sense of a smallest subobject from which the measure is induced by pushforward or in the Markov categories sense of support. In such a category, I would expect the equivalence classes of probability measures on $[0,1]$ with respect to absolute continuity to form part of the subobject poset of $[0,1]$.

I currently think that the opposite of the category of commutative $W^*$-algebras is an excellent candidate for such a category. And it has a number of other desirable features: completeness, cocompleteness, apparently cartesian closure (the latter per abelianizing Kornell's result on the monoidal closure of the opposite of the category of not-necessarily-commutative $W^*$-algebras), and an immediate generalization to the quantum case given by dropping commutativity. But actually this belongs to the "open problems" thread.