= Mecke formula for a Poisson random measure
{c}
{title2=$\mathbb E\sum_{x\in\Xi}H(x,\Xi-\delta_x)=\int\mathbb EH(x,\Xi)\,\rho(dx)$}
= Slivnyak-Mecke formula
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{synonym}
For nonnegative <measurable> $H$ and a <Poisson random measure> of sigma-finite intensity $\rho$, removing the selected point from a sum leaves an <independent> copy of the original process under the integral. For finite intensity, condition on the Poisson count, represent the points as <independent> samples from normalized $\rho$, and shift the count index using $n/n!=1/(n-1)!$. For sigma-finite intensity, first select points in a finite-intensity set, condition on the <independent> outside process and then exhaust the space. Repetition gives the corresponding formula for ordered distinct tuples of points.
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