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I must confess I hardly know anything about probability theory. Still, I'm interested in the following: Much like point-free topology, where one basically replaces topological spaces by their locales of open sets, I figured there is a way to do something similar with $\sigma$-algebras and with probability spaces.

Any thoughts on that? Does somebody know, whether this has been studied before?

Here are some more thoughts: I suppose a problem is how to recover the sample space $\Omega$ from a point-free probability space, as there is a no guarantee that there is an injection $\Omega \to \sigma$ from the sample space to the $\sigma$-algebra of a probability space. I wonder, how important it is to have a sample space at all. I (think I) know, that probability theory is actually about random variables, but do we really need a sample space to talk about those? Also, considering that there is no obvious notion of a morphism between probability spaces, maybe there are other objects we should look at?

(I asked this question on MSE and a user suggested to ask this on MO. I studied mathematics for about a year in university, so my background is not actually that sophisticated. I only know a little bit category theory, analysis and linear algebra)

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    $\begingroup$ It is good etiquette to wait a reasonable amount of time before cross-posting from MSE to here (say a week, you didn't even wait for an hour) and to cross-link the question, so as to avoid double efforts. You already had a reply on MSE, by the way. $\endgroup$ – Marco Golla Aug 29 '15 at 10:42
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    $\begingroup$ The standard point-free version of measure theory is to replace the algebraa of measurable sets by an abstract boolean algebra and the measure by a suitable function thereon. A good place to read about this and its motivation is the series of books by Fremlin, many of which are readily available online. $\endgroup$ – priel Aug 29 '15 at 11:00
  • $\begingroup$ @priel I assume you are referring to volume 3. It is definitely interesting and worth looking into, but it's measure theory not exactly probability theory. I was under the impression, that concepts like "random variables" are special to probability theory. Still, thank you for this information. There seem to be many useful ideas there. $\endgroup$ – Stefan Perko Aug 29 '15 at 11:28
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    $\begingroup$ There is also "Topological Riesz Spaces and Measure Theory" which discusses the classical spaces of random variables in this context. $\endgroup$ – priel Aug 29 '15 at 11:44
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    $\begingroup$ @Marco Golla, the poster's misbehavior is my fault, as I recommended him to post here. Also, the answer on MSE is quite unrelated to the question. Stefan Perko, asking a question was not a bad idea. There are many people (including me) who are also interested in an answer. $\endgroup$ – zhoraster Aug 29 '15 at 12:05
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Point-free probability theory, is treated in Kappos's book: Probability algebras and Stochastic spaces. Academic Press, 1969.

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I do not think the accepted answer is a complete one. To be honest there is no such a pointless theory as far as I know.

And I actually have read the book [Kappos] which could be viewed as a continution/smaller version of [Grenander](The dates is earlier also). The idea of using Markov transition kernel as morphisms between spaces is not quite extendible as we could see later in [Cencov]'s comprehensive treatment. As discussed in [Rota] a probability theory that is pointless is not available at the time he wrote down the paper (1998 Fubini talk), and as far as I concern the pointless notion that makes use of a locale is not well addressed in terms of "stochastic spaces". They provided algebraic structures but never a complete formal category definitions and their applications are rare if any.

Another attempt in this direction is to study the stochastic processes as a geometric object directly, which I think is more productive than the pure algebraic way. This approach dates back to the H.Cramer's approach of treating stochastic processes as a curve in Hilbert space.

The point of proposing a pointless probability theory is to discover some properties that are not clear when atoms/points are involved (yes it is also of categorical theoretic interest as well...but less). Since the geometric feature is revealed pretty well by using diffeomorphism flows over a space/group, the pointless theory itself attracts less interest now. (That is how I feel)

Reference

[Kappos]Kappos, Demetrios A. Probability algebras and stochastic spaces. Vol. 7. Academic Press, 2014.

[Cencov]Cencov, Nikolai Nikolaevich. Statistical decision rules and optimal inference. No. 53. American Mathematical Soc., 2000.

[Rota]Rota, G-C. "Twelve problems in probability no one likes to bring up." Algebraic combinatorics and computer science. Springer Milan, 2001. 57-93.

[Grenander]Grenander, Ulf. Probabilities on algebraic structures. Courier Corporation, 2008.

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    $\begingroup$ "I do not think the accepted answer is a complete one." - I did not expect anyone to give a 'complete' answer. I was okay with an answer like that after almost a year of this question being formally unanswered. $\endgroup$ – Stefan Perko Jul 24 '17 at 12:44
  • $\begingroup$ Regardless, thank you for your input as well. - Arguably, pointfree "something" is not an algebraic thing. It is actually dual to something algebraic (cf. the definition of the category of locales) and therefore rather geometric. Or so is my very rough understanding of this issue. $\endgroup$ – Stefan Perko Jul 24 '17 at 12:49
  • $\begingroup$ @StefanPerko As I understood the reference he pointed out is not even close to point-free theory that you want, Gerenander's book will be a better reference than that one. $\endgroup$ – Henry.L Jul 24 '17 at 12:51
  • $\begingroup$ Well, I don't have access to that book. From the ToC it seems it is more about replacing the boolean algebra ($\sigma$-algebra standin) with something else entirely. This may be interesting, but I myself right now don't know why. $\endgroup$ – Stefan Perko Jul 25 '17 at 19:07
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Possibly related... Caratheodory's book on measure and integration without points ...

Carathéodory, C. Mass und Integral und ihre Algebraisierung. (1956)

translated

Carathéodory, C. Algebraic theory of measure and integration. (1963) Translated from the German by F. E. J. Linton

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