Richard is right in #8. But the page needs a complete make-over. There’s a huge number of papers in this area. I wonder if we could entice someone like Tobias Fritz to help out.

]]>Thanks for your answers. The lambda seems to come from the Lambda calculus. (c. f. here: https://en.wikipedia.org/wiki/Lambda_calculus) and seems to be really a way to notate functions. To be honest, I never worked with Lambda calculus and do not know if it is common knowledge. Greetings, PJ

]]>The revision was written by David C, but the notation seems to come from the paper of Panangaden. I believe it is just an unnecessarily formal/complicated way to describe the function $x \mapsto h(x,B)$, i.e. the condition is that for all $B \in \Sigma_Y$, this function (which goes from $X$ to $\left[ 0, 1 \right]$) is bounded measurable.

]]>I don’t know, but just to note that the paragraph in question originates all the way from rev 1 back in May 2010 and seems to have been essentially untouched since.

]]>In “Probability theory” under nPOV view, one can read: $\forall x \in X . \lambda B \in \Sigma_Y . h(x, B)$ What does the lambda mean? Perhaps we should note what it is as it seems not to be the only widely accepted standard in probability theory textbooks. Greetings, PJ

]]>I gave the existing list of references a subsection “References – General” and then added below that “References – As Euclidean field theory”. There I added pointer to the recent:

Sourav Chatterjee,

*Yang-Mills for probabilists*, in:*Probability and Analysis in Interacting Physical Systems*, PROMS**283**(2019) Springer (arXiv:1803.01950, doi:10.1007/978-3-030-15338-0)Sourav Chatterjee,

*A probabilistic mechanism for quark confinement*, Comm. Math. Phys. 2020 (arXiv:2006.16229)

Of course, many more pointers must go here, to do this justice. But I leave it at that for the moment

]]>added pointer to

- Alex Simpson,
*Probability sheaves and the Giry monad*, 2017 (pdf, talk recording)

Update outdated link

Anonymous

]]>Added the recent reference

- Tobias Fritz,
*A synthetic approach to Markov kernels, conditional independence, and theorems on sufficient statistics*, (arXiv:1908.07021)

Strangely, we don’t seem to have an nForum discussion for probability theory.

I added a reference there to

- Chris Heunen, Ohad Kammar, Sam Staton, Hongseok Yang,
*A Convenient Category for Higher-Order Probability Theory*, (arXiv:1701.02547)

It replaces the category of measurable spaces, which isn’t cartesian closed, with the category of quasi-Borel spaces, which is. As they point out in section IX, what they’re doing is working with concrete sheaves on an established category of spaces, rather like the move to diffeological spaces.

[Given the interest in topology around these parts at the moment, we hear of ’C-spaces’ as generalized topological spaces arising from a similar sheaf construction in C. Xu and M. Escardo, “A constructive model of uniform continuity,” in Proc. TLCA, 2013.]

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