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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeNov 21st 2013

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 20th 2014
• (edited May 20th 2014)

I noticed that Giry monad used to jump into some details without really saying what’s going on first. I have now

• added a minimum of an Idea-section;

• split the “Outline” section into a Definition-section (and tried to make it read more like an actual definition) and a “Properties – Algebras”-section (which also deserves some streamlining, but I haven’t touched this);

• touched the formatting of some of the references.

Experts who care might want to polish this entry a bit more.

• CommentRowNumber3.
• CommentAuthorThomas Holder
• CommentTimeJun 24th 2014
Kirk Sturtz has put 'The Giry monad as a codensity monad' ([pdf](http://arxiv.org/pdf/1406.6030.pdf)) on the arXiv recently which I've added to reference section together with a link to Brendan Fong's paper on Bayesian networks. I guess Sturtz' results should be worked into the main text as well. There's also an unfortunate orthographic divergence occurring: the monad is called 'Giry monad' as well as 'Giry's monad' in the entry (title).
• CommentRowNumber4.
• CommentAuthorThomas Holder
• CommentTimeOct 17th 2014

I’ve added a reference to Tom Avery’s paper arxiv:1410.4432 and renamed the entry to ’Giry monad’ from ’Giry’s monad’.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeJul 25th 2016

Some of the literature on the Giry monad seems to be behind a paywall. Can anyone tell me whether the weak topology on the space $P(X)$ of Borel probability measures is the same as the topology induced by the Prokhorov metric? I have just added the latter as an example to further examples at Polish space, and would like to add this material plus a reference to Giry monad as well, but would like to check up on that point first.

1. Yes, at least for a complete separable metric space (that is to say Polish). Reference: Dudley, real analysis and probability.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeJul 25th 2016

Perfect; thank you Daniel. Is there a Theorem or page number where the statement is made?

2. complete answer: The statement holds actually even for any separable metric space. Reference: Dudley, Real Analysis and Probability, 2002, Theorem 11.3.3 page 395. Note that Dudley defines Prokhorov metric not according to normal terminology but introduces it calling it simply $\rho$. Anyway the cited theorem states the equivalence of all these notions of convergence, to roughly summarize the theorem: (almost) all reasonable notions of convergence of laws coincide on separable metric spaces (If you add even complete, I could not think of any reasonable notion that would not be equivalent to weak convergence).

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeJul 27th 2016

Thank you again, Daniel. This is very helpful indeed.

3. Added a short history of the Giry monad, and updated reference to my published work.

K. Sturtz

• CommentRowNumber11.
• CommentAuthorTim_Porter
• CommentTimeOct 31st 2018

I added in the first name of Elisabeth Burroni.

• CommentRowNumber12.
• CommentAuthorEvan Patterson
• CommentTimeFeb 8th 2019

I’m curious about the work by Voevodsky mentioned in this page. Unfortunately, I don’t speak Russian so I can’t watch the Moscow lecture. Does anyone know of any English-language paper, notes, or recorded lecture on this work?

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 8th 2019

Hi Evan. I looked around at the time that was added, but I never found anything.

• CommentRowNumber14.
• CommentAuthorAlizter
• CommentTimeFeb 8th 2019

Perhaps try contacting Zoran Skoda who wrote the part about Voevodsky.

4. Somebody at the Miami lecture must have taken notes. Could we try contacting somebody there? Looking at the faculty list, it is not obvious who might have invited him/known him, though.

• CommentRowNumber16.
• CommentAuthorAlizter
• CommentTimeFeb 9th 2019

Perhaps a good start is contacting Daniel Grayson?

• CommentRowNumber17.
• CommentAuthorspitters
• CommentTimeFeb 9th 2019
• (edited Feb 9th 2019)
Vladimir send me a copy in 2014. Dan Grayson has now made it available [here](http://www.math.ias.edu/Voevodsky/voevodsky-publications_abstracts.html#markov)
• CommentRowNumber18.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 9th 2019

You need to choose Markdown+Itex, so

Dan Grayson has now made it available here

• CommentRowNumber19.
• CommentAuthorEvan Patterson
• CommentTimeFeb 9th 2019

Very nice. Thanks everyone for your help!

• CommentRowNumber20.
• CommentAuthorEvan Patterson
• CommentTimeFeb 9th 2019