This post requires some knowledge of measure theory.
Today I’m going to show that the product of two measurable spaces and , is actually the product in the category of measurable spaces. See Product (category theory).
The category of measurable spaces, , is the collection of objects , where is a set and is a -algebra on , and the collection of morphisms such that
- for all .
Such a function is called a measurable morphism, and is called a measurable space.
Given two measurable spaces and we can define their product , where is the cartesian product of and , and is the -algebra generated by the sets of the form and with and . We will need the following definition: A -algebra on a set is said to be coarser than a -algebra on if . In exercise 18 of Terry Tao’s notes on product measures, it is shown that is the coarsest -algebra on such that the projection maps and are both measurable morphisms.
Now, we are finally ready to show that is the categorical product of the measurable spaces and . If and are measurable morphisms, then we need to show that there exists a unique measurable morphism such that and . Because the cartesian product is the product in the category of sets, we only have one choice for such a map. Indeed, .
We claim that is measurable. Indeed, because the pullback respects arbitrary unions and complements, and , we only need to show that and for all and (see remark 4 of Terry Tao’s notes on abstract measure spaces). This is easy to show:
are both measurable because and .
Thus, we have shown that is actually the product of the measurable spaces and in the category of measurable spaces. This is reassuring, otherwise the term “product space” would be misleading.
I would like to know if there is a category of measure spaces with objects , where is a set, is a -algebra on , and is a measure. There would need to be some extra condition on morphisms in this category, otherwise we couldn’t distinguish between triples and where . If this was a category, I don’t believe products could exist. Indeed, if the measure spaces and are not -finite, then there can be multiple measures on : see Terry Tao’s notes on product measures.
Another question is whether equalizers, coproducts, coequalizer, etc. exist in the category of measurable spaces. If a sufficient number of these properties exist, then we can take categorical (co)limits of measurable spaces. These might be of some interest already, but I have not looked into it.