Category for measure spaces? I know some things about measures/probabilities and I know some things about categories. Shortly I realized that uptil now I have never encountered something as a category of measure spaces. It seems quite likely to me that something like that can be constructed. I am an amateur however and my scope is small. I have two questions:

1 Is there indeed material of this sort and can you tell me about it? The whereabouts for instance.
2 Is there a reason for the fact that uptil now I did not find anything of the sort? Is it indeed rare for some reason?

 A: I'm just going to record some references that you might want to take a look at. They might be a little too heavy on the category theory for your taste, but they also might provide further places to look. 


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*$n$-Category Café post: Category Theoretic Probability Theory 

*Related in content, the nLab article on probability theory; see especially the section "Probability theory from the nPOV" 

*The nLab article on measurable spaces. See especially the connection with von Neumann algebras, where it is localizable measurable spaces which are the pertinent concept for the appropriate Gelfand-Naimark duality, and see also references to posts by Dmitri Pavlov at MathOverflow. 
A: This has been asked before:


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*Is there an introduction to probability theory from a structuralist/categorical perspective?
And for the notion of product:


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*Can one view the Independent Product in Probability categorially?

*Is there a category structure one can place on measure spaces so that category-theoretic products exist?
