One can construct a category of probability spaces, but this [category has no products](https://mathoverflow.net/questions/49426/is-there-a-category-structure-one-can-place-on-measure-spaces-so-that-category-th). Now probability theory relies strongly on the ability to build independent products, the product measure. In a sense, the notion of independence is what distinguishes probablity theory from the theory of finite measures.



> 
> Is there a categorial way to make
>  sense of and enlighten the notion of independent
>  products in category theory?
> 
> 
> 


It is possible to formulate independence in Lawvere's category of *probabilistic mappings* (Borel spaces as objects and Markov kernels as morphisms) in terms of constant morphisms, but I think this is not very enlightening, conditional independence is built into the morphisms. Maybe, this is what one has to do when putting probability center stage?


I do know the rudiments of categry theory, but I would prefer an answer that does not require too much immersion in category thory, provided that is possible. 



Can one view the Independent Product in Probability categorially?
Very nice question! ;) I wrote a short paper about this question about ten years ago, see <http://arxiv.org/abs/math/0206017> (My apologies for advertising my own work, but this is exactly the question I asked myself at that time).


The product of probability spaces is tensor product in the sense of category, as Martin Brandeburg also pointed out in a comment. But it has an additional structure, you have natural morphisms onto the factors in the tensor product. This is because the projections onto the two factors that you have for the Cartesian product (of sets) preserve the measures, so they are also morphisms in the category of probability spaces. I called this structure a **tensor product with projections**: for two objects $\Omega\_i=(\Omega\_i,\mathcal{F}\_i,P\_i)$, $i=1,2$, you get $\Omega\_1\otimes\Omega\_2=(\Omega\_1\times\Omega\_2,\mathcal{F}\_1\otimes\mathcal{F}\_2,P\_1\otimes P\_2)$ **and** random variables $X\_i:\Omega\_1\otimes\Omega\_2\to \Omega\_i$, $i=1,2$.


You can use this "tensor product with projections" to characterise independence of random variables: two r.v. $Y\_i:\Omega\to\Omega\_i$, $i=1,2$, defined on the same probability space $\Omega$, are independent iff they factorise, i.e., if there exists a r.v. $Z:\Omega\to\Omega\_1\otimes \Omega\_2$ such that $Y\_i=X\_i\circ Z$, $i=1,2$.


The notion dualises to the algebras of functions on a probablity space, where it becomes a **tensor product with inclusions**. Generalising to not necessarily commutative algebras, it includes notions of independence used in noncommutative (or quantum) probability, like the freeness.


