L2 construction of compensated Poisson integrals ($\mathbb{E}{\left|\int f(\mu -\nu )\right|}^2=\int |f{|}^{2}d\nu$)

For deterministic square-integrable $f$ and a Poisson random measure of intensity $\nu$, first define the integral for simple functions on disjoint finite-intensity sets. Independence and the equality of Poisson distribution mean and variance prove the displayed isometry. Completion defines the integral for all $f\in L^2(\nu )$, even when $\int |f|d\nu =\infty$. A small-jump compensated martingale is thereby defined through convergence in Lp, not by subtracting two divergent ordinary integrals.