Is the product of $\sigma$-algebras a tensor product in some sense?

In the wikipedia page for product measures it says:

Let $(X_{1},\Sigma _{1})$ and $(X_{2},\Sigma _{2})$ be two measurable spaces, that is, $\Sigma _{1}$ and $\Sigma _{2}$ are sigma algebras on $X_{1}$ and $X_{2}$ respectively, and let $\mu _{1}$ and $\mu _{2}$ be measures on these spaces. Denote by $\Sigma _{1}\otimes \Sigma _{2}$ the sigma algebra on the Cartesian product $X_{1}\times X_{2}$ generated by subsets of the form $B_{1}\times B_{2}$, where $B_{1}\in \Sigma _{1}$ and $B_{2}\in \Sigma _{2}$. This sigma algebra is called the tensor-product σ-algebra on the product space. A product measure $\mu _{1}\times \mu _{2}$ is defined to be a measure on the measurable space $(X_{1}\times X_{2},\Sigma _{1}\otimes \Sigma _{2})$ satisfying the property:

$(\mu _{1}\times \mu _{2})(B_{1}\times B_{2})=\mu _{1}(B_{1})\mu_{2}(B_{2})$

for all $B_{1}\in \Sigma _{1},\ B_{2}\in \Sigma _{2}$.

a. I can see that the product of $\sigma$-algebras is a categorical product; but is it correct to say that it is also a tensor product in some analogue to the tensor product in vector spaces? In which case how? And is there a universal property that expresses it as such?

b. I don't see how to express the product of measures as a categorical product. Is it?

Though tensor products are usually defined for vector spaces, its most generally defined for abelian groups:

For $A$ and $B$ two abelian groups, their tensor product $A\otimes B$ is a new abelian group which is such that a group homomorphism $A\otimes B \rightarrow C$ is equivalently a bilinear map out of $A$ and $B$.

Then some evidence that we might be able to interpret the product of measures as a kind of 'tensor product' is that supposing $B_1$ and $B_1'$ are disjoint in $\Sigma_1$, we have:

$(\mu _{1}\times \mu _{2})((B_{1} \sqcup B_1')\times B_{2})=\mu _{1}(B_{1} \sqcup B_1')\mu_{2}(B_{2})=(\mu _{1}B_{1}+ \mu _{1}B_{1}') \times \mu_{2}B_{2}$

=$(\mu_1 \times \mu_2)(B_1,B_2)+(\mu_1 \times \mu_2)(B_1',B_2)$

That is it satisfies the bilinearity property for the tensor product for disjoint sets.

Solution 1:

In my opinion this should simply be called the product of the two measurable spaces. In fact, it satisfies the corresponding universal property.

To answer your general question, you should look at monoidal categories. Every category with products has a monoidal structure with $\otimes = \times$. The category of vector spaces is also monoidal with $\otimes$ being the usual tensor product.

Solution 2:

To have a universal property one first must define a category. So, if we take the category of measurable spaces, where objects are pairs $(X,\Sigma)$ of a set and a sigma algebra on it, and morphisms $f:(X,\Sigma)\to (Y,\Sigma ')$ are functions $f:X\to Y$ which are measurable, then the tensor product you mention is nothing but the categorical product. Notice however that your question mentioned measures $\mu_1,\mu_2$ which are never used. That may indicate that what you actually had in mind was to talk about the product of the measures in some categorical way. That is much more subtle since it is entirely not clear what is a useful way to define a category of measures. Intuitively, it should have objects triples $(X,\Sigma, \mu)$, a set with a sigma algebra on it, and a measure as well. But what should the morphisms be?

There are various ways to answer that question, leading to interesting results, of which I will only mention this article.