 Laplace functional of a Poisson random measure

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 $f={\sum }_j{u}_j{1}_{A_j}$ for disjoint finite-intensity sets gives the joint Laplace transform of independent Poisson counts, so characterizes the law of a Poisson random measure.