Campbell's theorem

For a Poisson point process $\Pi$ with intensity measure $\mu$ and a square-integrable measurable function $g$,The second identity follows because distinct points contribute the square of the mean, while the diagonal contribution is $\int g^{2}d\mu$.

The expectation identity holds for a general point process with intensity measure $\mu$ whenever the sum is integrable. The stated variance identity uses the additional Poisson point process assumption.

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.