I'm studying a proof and have problems with a step where a probability is converted into a Lebesgue Stieltjes integral with respect to a distribution function.
Let $(X_i)$ be a sequence of independent, identically distributed random variables with distribution function $F$, $S_n = \sum_{i=1}^n X_i$ and let $$\Psi_n(a) = \mathbb P(\max_{1\leq k\leq n}S_k > a).$$
I'm trying to understand the equality: $$\mathbb P(X_1\leq a, X_1 + \max_{2\leq k\leq n+1} \sum_{i=2}^k X_i > a) = \int_{(-\infty, a]} \Psi_n(a-x) \mathbb d F(x) $$
I can appreciate intuitively what's happening here: Since the $X_i$ are i.i.d. we can shift indices without affecting the distribution of the sum. Then we condition on $X_1 = x$ in some sense and then integrate over all possible values where the limits of integration are given by the event $\{X_1 \leq a\}$. But I don't really have a rigorous understanding of what's really happening here. Can someone help me with that?
The context is an induction proof of the Lundberg inequality for the Sparre Andersen model from ruin theory.