One can construct a category of probability spaces, but this [category has no products][1]. 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. 


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