Currently I'm reading a paper about model uncertainty and Value-at-Risk aggregation. The goal is to calculate boundaries for $VaR_\alpha(X+Y)=F^{-1}_{X+Y}(\alpha)=inf\{m|\mathbb{P}(X+Y+m<0)\leq\alpha\}$ with unknown dependency structure, so we don't know the joint distribution of (X,Y).
My question is why would they focus on the joint distribution of (X,Y) but not on the distribution of X+Y to calculate the boundaries $\underline{VaR}_\alpha(X+Y)=inf\{VaR_\alpha(X+Y):F_{(X,Y)}$ is element of the class of possible joint distributions$\}$ and $\overline{VaR}_\alpha(X+Y)$ analogous
Edit: I think I got the answer by applying $\int g(X,Y)d\mathbb{P}=\int g(x,y)d\mathbb{P}_{X,Y}$, so for continuous functions, but actually I'm not too sure if this equality is right.