This should be simple, and with a bit of study I should get there, but I am very tired and in a need for a hand. Thanks in advance.
If $\rho$ is a probability measure over $Y$. For each $y \in Y$ there is a measure $\mu_y$ over $X$, (you can assume $X=Y=[0,1]$). If I wish to define the unconditional measure $\mu$ over $X$, $\mu(A)=\int _{y \in Y} \mu_y(A) \partial \rho$, for any $A \in \sigma_X$ ( $\sigma$-algebra can be Borel). When is $\mu$ measurable? Do I need any special condition on $\mu_y$?
If $\rho^n$ converges weakly to $\rho$, $\rho^n \Rightarrow \rho$ and $\mu_y^n \Rightarrow \mu_y$ for all $y \in Y$, does $\mu^n \Rightarrow \mu$?