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.

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The usual category of measure spaces requires the morphisms to be measure-preserving, i.e. $\phi_* \mu_X = \mu_Y$. If so, I don’t think there are any category-theoretic products, even if you restrict to finite measure spaces (or even discrete measure spaces of finite cardinality). The basic reason is that given two measure spaces X and Y, there are multiple measures one can place on whose marginal distributions (i.e pushforwards) on X and Y are and . From a probability perspective, this reflects the fact that the distribution of individual variables X, Y is not enough to determine the joint distribution (X,Y), because the two variables are not necessarily independent.

However there may be some other category structure one can place on measure spaces which allows for category-theoretic products. You might try asking the question on MathOverflow.

Another obstacle is that the canonical projection map might not be measure preserving. A simple example is the the product of the Borel measures on . The pushforward measure of the unit interval is .

I asked this question on mathoverflow:

http://mathoverflow.net/questions/49426/is-there-a-category-structure-one-can-place-on-measure-spaces-so-that-category-th

Damek

Reblogged this on Being simple.