Existence of a Markov category of measurable spaces with conditional expectations

A Markov category has objects given by measurable spaces $(X,\Sigma_X)$, or variants such as measure algebras, and morphisms $(X,\Sigma_X) \to (Y,\Sigma_Y)$ given by suitable positive unital linear maps

$$\mathcal{L}^\infty(Y,\Sigma_Y) \longrightarrow \mathcal{L}^\infty(X,\Sigma_X),$$

that preserve bounded directed suprema. Its monoidal structure is a suitable variant of the product of measurable spaces, and the comultiplication maps are induced by restriction to the diagonal

$$\mathcal{L}^\infty(X\times X,\Sigma_X \otimes \Sigma_X) \longrightarrow \mathcal{L}^\infty(X,\Sigma_X).$$

Conditional-expectation Markov category conjecture. Such a Markov category exists, with conditionals in the sense of the paper's definition, and these conditionals are given by conditional expectations. This would provide a categorical framework for conditional expectations in measure-theoretic probability. The precise choice of measurable-space variants, morphisms, and product structure remains to be established.