Is the Product of Measurable Spaces the Categorical Product?

This post requires some knowledge of measure theory.

Today I’m going to show that the product of two measurable spaces {(X, \mathcal{B}_X)} and {(Y,  \mathcal{B}_Y)}, is actually the product in the category of measurable spaces. See Product (category theory).

The category of measurable spaces, {\mathbf{Measble}}, is the collection of objects {(X,\mathcal{B}_X)}, where {X} is a set and {\mathcal{B}_X} is a {\sigma}-algebra on {X}, and the collection of morphisms {\phi : (X, \mathcal{B}_X) \rightarrow  (Y, \mathcal{B}_Y)} such that

  1. {\phi : X \rightarrow Y};
  2. {\phi^{-1}(E) \in \mathcal{B}_X} for all {E \in \mathcal{B}_Y}.

Such a function {\phi} is called a measurable morphism, and {(X, \mathcal{B}_X)} is called a measurable space.

Given two measurable spaces {(X, \mathcal{B}_X)} and {(Y, \mathcal{B}_Y)} we can define their product {(X\times Y, \mathcal{B}_X \times  \mathcal{B}_Y)}, where {X\times Y} is the cartesian product of {X} and {Y}, and {\mathcal{B}_X \times  \mathcal{B}_Y} is the {\sigma}-algebra generated by the sets of the form {E \times Y} and {X \times F} with {E \in \mathcal{B}_X} and {F \in  \mathcal{B}_Y}. We will need the following definition: A {\sigma}-algebra {\mathcal{B}} on a set {Z} is said to be coarser than a {\sigma}-algebra {\mathcal{B}'} on {Z} if {\mathcal{B} \subseteq \mathcal{B}'}. In exercise 18 of Terry Tao’s notes on product measures, it is shown that {\mathcal{B}_X \times  \mathcal{B}_Y} is the coarsest {\sigma}-algebra on {X\times Y} such that the projection maps {\pi_X} and {\pi_Y} are both measurable morphisms.

Now, we are finally ready to show that {(X\times Y,  \mathcal{B}_X\times \mathcal{B}_Y)} is the categorical product of the measurable spaces {(X,  \mathcal{B}_X)} and {(Y,  \mathcal{B}_Y)}. If {\phi_X : (Z, \mathcal{B}_Z)  \rightarrow (X, \mathcal{B}_X)} and {\phi_Y : (Z,  \mathcal{B}_Z) \rightarrow (Y, \mathcal{B}_Y)} are measurable morphisms, then we need to show that there exists a unique measurable morphism {\phi_{X\times Y} : (Z, \mathcal{B}_Z)  \rightarrow (X\times Y, \mathcal{B}_X \times  \mathcal{B}_Y)} such that {\phi_X = \pi_X \circ  \phi_{X\times Y}} and {\phi_Y = \pi_Y \circ  \phi_{X\times Y}}. Because the cartesian product is the product in the category of sets, we only have one choice for such a map. Indeed, {\phi_{X \times Y} = (\phi_X, \phi_Y)}.

We claim that {\phi_{X \times Y}} is measurable. Indeed, because the pullback {\phi_{X \times Y}^{-1} : 2^{X\times  Y} \rightarrow 2^Z} respects arbitrary unions and complements, and {\phi_{X\times Y}(\emptyset) =  \emptyset}, we only need to show that {\phi_{X\times Y}^{-1}(E \times Y) \in \mathcal{B}_Z} and {\phi_{X\times Y}^{-1}(X \times F) \in  \mathcal{B}_Z} for all {E \in  \mathcal{B}_X} and {F \in  \mathcal{B}_Y} (see remark 4 of Terry Tao’s notes on abstract measure spaces). This is easy to show:

\displaystyle  \begin{array}{rcl}  \phi^{-1}_{X\times Y} (E\times Y)  &=& (\pi_X \circ \phi_{X\times Y})^{-1}(E) \\ &=&  \phi_X^{-1}(E) \\ \phi^{-1}_{X\times Y} (X \times F) &=& (\pi_Y  \circ \phi_{X\times Y})^{-1}(F) \\ &=& \phi_Y^{-1}(F)  \end{array}

are both {\mathcal{B}_Z} measurable because {E \in \mathcal{B}_X} and {F \in  \mathcal{B}_Y}.

Thus, we have shown that {(X \times Y, \mathcal{B}_X\times  \mathcal{B}_Y)} is actually the product of the measurable spaces {(X, \mathcal{B}_X)} and {(Y,  \mathcal{B}_Y)} 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 {(X, \mathcal{B}_X, \mu_X)}, where {X} is a set, {\mathcal{B}_X} is a {\sigma}-algebra on {X}, and {\mu_X : \mathcal{B}_X \rightarrow [0, +\infty]} is a measure. There would need to be some extra condition on morphisms in this category, otherwise we couldn’t distinguish between triples {(X, \mathcal{B}_X, \mu_X)} and {(X,  \mathcal{B}_X, \mu_X')} where {\mu_X \neq  \mu_X'}. If this was a category, I don’t believe products could exist. Indeed, if the measure spaces {(X, \mathcal{B}_X,  \mu_X)} and {(Y, \mathcal{B}_Y,  \mu_Y)} are not {\sigma}-finite, then there can be multiple measures on {(X \times Y,  \mathcal{B}_X \times \mathcal{B}_Y)}: 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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3 thoughts on “Is the Product of Measurable Spaces the Categorical Product?

  1. The usual category of measure spaces requires the morphisms \phi: X \to Y 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 X \times Y whose marginal distributions (i.e pushforwards) on X and Y are \mu_X and \mu_Y. 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.

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