A Point-free probability theory?

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.

Point-free probability theory, is treated in Kappos's book: Probability algebras and Stochastic spaces. Academic Press, 1969.

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