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If I have two measure space $X,Y$, with measure $\mu_X$ and $\mu_Y$, the usual way to define a measure for the combined measure space $U=X \times Y$, is the product measure, i.e., $$\mu_U = \mu_X \times \mu_Y$$

However, this implies the independence between $X$ and $Y$, in my case, I want to define a measure that has dependency on each other. The corresponding idea would be define a conditional measure, $$ \mu_{U} = \mu_{X} \mu_{Y|X} $$ It is possible? Has any one heard about anything like it?

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Yes these are called Markov kernels. In this paper, on page 229, he also define the conditional probability using the Radon-Nikodym theorem.

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