Campbell first-moment formula ($\mathbb{E}{\sum }_{z\in \Pi }h(z)=\int hd\mu$)

For a Poisson random measure $\Pi$ with intensity $\mu$ and measurable nonnegative $h$, $\mathbb{E}\int hd\Pi =\int hd\mu$, allowing both sides to be infinite. For signed $h$, the same identity holds if $\int |h|d\mu <\infty$. This first-moment statement does not require square integrability. Applying it to a nonnegative volume mark gives the expected total volume in a marked Poisson model.