I would like your help to rewrite an expression involving the law of total probability with continuous random variables.
Consider the random variables $Y,V,T$ with supports $\mathcal{Y},\mathcal{V},\mathcal{T}$. Only $\mathcal{Y}$ is finite, the other two sets are non-finite.
What I want to do is to rewrite the probability distribution of $(Y,V)$ as a function of the probability distributions $P_V$, $P_{T}(\cdot| V)$, $P_{Y}(\cdot| T)$.
If $\mathcal{V}$ and $\mathcal{T}$ were finite, then we would have had that $\forall y\in \mathcal{Y}$ and $\forall v\in \mathcal{V}$
$$ P_{Y,V}(y,v)=P_V(v)\times \sum_{t\in \mathcal{T}} P_T(t|v) \times P_Y(y|t,v) $$
I'm unable to rewrite this expression when $\mathcal{V}$ and $\mathcal{T}$ are not finite. I don't want to use densities, but just cumulative distribution functions. Could you help?