= Compensated Poisson integral
For a <Poisson random measure> $N$ with sigma-finite <intensity measure> $\mu$, a real deterministic simple integrand $f=\sum_i a_i\mathbf1_{A_i}$ on disjoint sets of finite intensity has <compensated Poisson integral>
$$
I(f)=\sum_i a_i[N(A_i)-\mu(A_i)].
$$
Its <expected value> is zero. <Independence> of disjoint counts and the Poisson <variance> formula give $\mathbb E[I(f)^2]=\int f^2\,d\mu$.
This <isometry> extends the integral by completion to deterministic $f\in L^2(\mu)$. If $f$ is not also integrable, writing two separate finite integrals $\int f\,dN-\int f\,d\mu$ can be invalid; the compensated integral is defined through the joint centered limit. For a constant-rate <Poisson process>, taking $f=\mathbf1_{[0,t]}$ gives $N_t-\lambda t$ with <variance> $\lambda t$.<Lectures on the Poisson Process by Günter Last and others>[Section 12.1, compensated Wiener-Itô integral, isometry and L2 extension]{ref}
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