Naturality characterizes product measures and relative product measures

Source abstract

We show that the product measure is the only natural way to assign to each pair of probability measures on measurable spaces a probability measure on their product. Here, naturality is meant in the sense of category theory and amounts to the condition that the assignment commutes with pushforward along measurable maps. In the standard Borel setting, we also prove an analogous result for relative product measures over a fixed base probability space.