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Stream: learning: questions

Topic: monad for measures

Matteo Capucci (he/him) (Nov 03 2020 at 11:59):

We know the functor sending a measurable space $X$ to its measurable space of probability measures is a monad. Is this true also if we replace 'probability measures' with arbitrary measures?
Put in other words, is there a point in the construction in which $\mu(X)=1$ (or at least $\mu(X) < \infty$) is actually crucial? I'm too lazy to work it out now so I'm asking here to see if someone has a ready answer for me

John Baez (Nov 03 2020 at 16:48):

At first I thought finite measures were required. If T is your monad, you need a map TTX $\to$ TX, which converts a measure on the space of measures on X into a measure on X. We get this by integration. If we allow infinite measures, this integral could be infinite. But now I think that's fine: there's a measure on X that sends every nonempty set to $\infty$, for example, so a result like that is allowed.

Nathaniel Virgo (Nov 04 2020 at 01:15):

There was some discussion of this issue at theory: probability / improper priors. @Sam Staton linked to his paper where he uses a category of "s-finite" kernels. (s-finite measures are a particularly well-behaved class of possibly-infinite measures.)

I guess that's more about defining a Markov category of s-finite kernels directly, rather than as the Kleisli category of a monad, though.

Matteo Capucci (he/him) (Nov 04 2020 at 10:35):

@Nathaniel Virgo does s-finite mean $\sigma$-finite?

Nathaniel Virgo (Nov 04 2020 at 10:36):

No, it's different! https://en.wikipedia.org/wiki/S-finite_measure

Nathaniel Virgo (Nov 04 2020 at 10:37):

I'm not really an expert on this measure-theoretic stuff though - I'm hoping someone else will chime in

Matteo Capucci (he/him) (Nov 04 2020 at 14:23):

Uh that's a cool concept, I've never seen it before!

Matteo Capucci (he/him) (Nov 04 2020 at 14:25):

John Baez said:

At first I thought finite measures were required. If T is your monad, you need a map TTX $\to$ TX, which converts a measure on the space of measures on X into a measure on X. We get this by integration. If we allow infinite measures, this integral could be infinite. But now I think that's fine: there's a measure on X that sends every nonempty set to $\infty$, for example, so a result like that is allowed.

Btw this is also where I thought problem could arise, though as you point out... not really! Though I can't still swear that there are no problem convolving one measure over the other when they could be infinite.

Reid Barton (Nov 04 2020 at 14:29):

In Lean's mathlib library this monad is formalized without the restriction to probability (or bounded) measures.
https://github.com/leanprover-community/mathlib/blob/master/src/measure_theory/giry_monad.lean

Matteo Capucci (he/him) (Nov 04 2020 at 14:30):

Thanks Reid! I guess this settles the question

John Baez (Nov 04 2020 at 17:16):

Nice! A good general principle: $+\infty$ is not a problem when dealing with sums or integrals unless you also have negative numbers around.

John Baez (Nov 04 2020 at 17:17):

Dealing with signed possibly-infinite measures is much more tricky than dealing with possibly-infinite measures, because there's no good definition of $+\infty - \infty$.

Matteo Capucci (he/him) (Nov 04 2020 at 17:18):

Indeed

John Baez (Nov 04 2020 at 17:19):

But there's never an ambiguity when adding (possibly infinite, even uncountable) sets of numbers in $[0,+\infty]$.

John Baez (Nov 04 2020 at 17:24):

We can talk about "infinitary Lawvere theories" where we allow operations of arbitrary arity (though it's typical to bound the arity by some cardinal or other... take an inaccessible cardinal if you want).

John Baez (Nov 04 2020 at 17:26):

So, we can talk about an "infinitary commutative monoid", in which sums of arbitrary cardinality are well-defined, and infinitary versions of the associative and commutative laws hold.

John Baez (Nov 04 2020 at 17:26):

Then $\mathbb{N} \cup \{+\infty\}$ is an infinitary commutative monoid, and so is $[0,\infty]$.

John Baez (Nov 04 2020 at 17:26):

Measure theory uses the latter.

Matteo Capucci (he/him) (Nov 04 2020 at 17:50):

Wow this is cool!

John Baez (Nov 04 2020 at 18:35):

Thanks! Here's an important and fun fact: if you add infinitely many numbers in $[0,\infty]$ and get a finite answer, at most countably many of these numbers can be nonzero.

sarahzrf (Nov 05 2020 at 00:24):

then what do you call integration? :thinking:

sarahzrf (Nov 05 2020 at 00:24):

jk

Nathaniel Virgo (Nov 05 2020 at 00:27):

John's fact is why we need measure theory in the first place. It means that if you try to assign probabilities to elements of a set directly, only countably many can ever be nonzero

Dan Doel (Nov 05 2020 at 01:32):

@sarahzrf Infinitessimals are non-nonzero, I think.

sarahzrf (Nov 05 2020 at 01:52):

nice :)

sarahzrf (Nov 05 2020 at 01:52):

read my mind