Solution
= Solution
Campbell's formula for a <Poisson point process> gives
$$
\mathbb ES=\int_{\mathbb R^d}g(u)\lambda(u)du.
$$
Expanding $S^2$ separates equal and distinct Poisson points. The factorial-moment formula gives
$$
\mathbb ES^2
=\int g(u)^2\lambda(u)du
+\left(\int g(u)\lambda(u)du\right)^2.
$$
Subtracting the square of the mean yields
$$
\boxed{\operatorname{var}S=\int_{\mathbb R^d}g(u)^2\lambda(u)du.}
$$