Well-behaved Markov category with arbitrary products

Determine a well-behaved Markov category for measure-theoretic probability that admits products of arbitrary families of objects (i.e., uncountable products in the sense of Kolmogorov products).

Background

The standard Borel Markov category supports many probabilistic constructions but fails to be closed under uncountable products. The authors aim for a setting where measure-theoretic probability admits arbitrary products while retaining key categorical properties such as the existence of conditionals.