Let $X, Y$ be random variables on a measure space $(\Omega, \mathcal{A}, \mathbb{P})$. I'd like to write the cdf $F_{X+Y}$ of $X + Y$ in terms of the cdfs $F_X$ and $F_Y$.
This is my approach
$$F_{X+Y}(z) = \mathbb{P}(X + Y \leq z) = \mathbb{E}[1_{\{X + Y \leq z\}}] = \int 1_{\{X + Y \leq z\}} \,d\mathbb{P}$$
How can I proceed at this point?