Let $F(x)$ denote the CDF of a random variable. We have for $0 < a < 1$
$$ F(x) = \begin{cases} 0 & \text{ if } x < 0 \\ a x & \text{ if } 0 \leq x < 1 \\ 1 & \text{ if } x \geq 1 \end{cases} $$
In particular, $F$ has a point mass at $1$. Now, I would like to compute the CDF of 2 draws from $F$. For $s \geq 0$,
$$ F_2(s) = P[X_1 + X_2 \leq s] = \int_0^1 P[X_1 + X_2 \leq s | X_2 = t] dP[X_2 \leq t] = \int_0^1 F(s-t) dF(t) $$
Does my integral exist? A popular existence theorem requires for $f$ to be integrable on $\alpha$, that both have finitely many discontinuities, and $f$ has to be continuous whenever $\alpha$ is not, and vice-versa.
Now, for $s = 2$, both $F(s-t)$ and $F(t)$ are discontinuous at $t=1$. Is there any existence theorem I can use to justify computing the R-S integral here, or how should I proceed?