Consider $X_{1} \dots $ i.r.v with $Exp(\lambda)$ distribution. Let's consider $S_n = X_{1} + \dots + X_{n}$.
So we may construct a point process $Y = \{Y(B) = \sum \mathbb{1}_{B}(S_n) , B\in \mathfrak{B}(\mathbb{R}) \}$.
No we need to find $\mathfrak{L}(f) = \mathbb{E}(\exp[-\sum_n f(S_n)])$,
My attempt :
First of all let's suppose that we have $\xi $~$Gamma(n,\frac{1}{\lambda})$ , when $\mathbb{E}(\exp[-s\xi]) = \left(\dfrac{\lambda}{\lambda + s}\right)^n$.
Let's consider $f = \sum_{k = 1}^{m} a_k\mathbb{1}_{B_k}$, hence we have :
$\mathfrak{L}(f) = \mathbb{E}\exp[-\sum_{n}^{\infty} \sum_k^{m} a_{k} \mathbb{1}_{B_k}(S_n)]$ replace sum sign due to positive terms.
$\mathfrak{L}(f) = \prod \mathbb{E} \exp[-a_k Y(B_{k})]$, but there is the stuck problem.
What should I do now?