Monads for measurable queries in probabilistic databases

We consider a bag (multiset) monad on the category of standard Borel spaces, and show that it gives a free measurable commutative monoid. Firstly, we show that a recent measurability result for probabilistic database queries (Grohe and Lindner, ICDT 2020) follows quickly from the fact that queries can be expressed in monad-based terms. We also extend this measurability result to a fuller query language. Secondly, we discuss a distributive law between probability and bag monads, and we illustrate that this is useful for generating probabilistic databases