= Laplace functional of a Poisson random measure
{c}
{title2=$\mathbb E e^{-M(f)}=\exp(-\int(1-e^{-f})\,d\mu)$}
For nonnegative measurable $f$, <independent> Poisson counts first prove this formula for simple functions on disjoint <measurable sets>. Monotone approximation gives the general formula, allowing infinite integrals with $e^{-\infty}=0$. Conversely, the formula evaluated at $f=\sum_ju_j\mathbf1_{A_j}$ for disjoint finite-intensity sets gives the joint <Laplace transform> of <independent> Poisson counts, so characterizes the law of a <Poisson random measure>.
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