 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.