I found the following expression in a paper but I'm not quite sure where does it come from:
let $Y$ and $V$ be two random variables defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and with support respectively $\mathcal{Y}$ and $\mathcal{V}$, both compact sets. Let $M \subset \mathcal{Y}\times \mathcal{V}$. Hence, $$ \mathbb{P}((Y,V) \in M)=\int_{\mathcal{t \in V}} \mathbb{P}(Y \in M_t| V=t)\mathbb{P}_V(dt) \hspace{1cm}\text{(*)} $$ where $M_t:=\{k \in \mathcal{Y}:(k,t) \in M \text{ $k$ singleton} \}$. I don't understand equation (*) and, in particular, what $\mathbb{P}_V(dt)$ is.