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 Iｆｅｅｌ)

[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.

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.