Spatial window average

Let $(W_n)$ be a Van Hove sequence in ${\mathbb{R}}^d$ as in Van Hove sequences of spatial windows, under its ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption. The Borel-measurability result makes each $W_n$ Lebesgue measurable, and the complete-measure result supplies the Lebesgue measure used below. On a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ (Probability measures and probability spaces), let $(X(s){)}_{s\in {\mathbb{R}}^d}$ be a real random field in the coordinatewise sense of Random fields, stationarity, covariance, and variograms; its coordinates are real random variables (Random elements and real random variables) such that the map $(s,\omega )\mapsto X(s,\omega )$ is measurable for $\mathcal{B}({\mathbb{R}}^d)\otimes \mathcal{F}$ (The Borel sigma-algebra of a topological space). For $W_n$, assume that ${\int }_{W_n}\mathrm{\mid }X(s,\omega )\mathrm{\mid }ds<\mathrm{\infty }$ for almost every $\omega$. Define the spatial window average on that full-measure event by ${\bar{X}}_{W_n}(\omega ):=\frac{1}{\mathrm{\mid }{W}_n\mathrm{\mid }}{\int }_{W_n}X(s,\omega )ds,$ and set it to $0$ on the measurable exceptional null set. Here $\mathrm{\mid }{W}_n\mathrm{\mid }$ and the integral use $d$-dimensional Lebesgue measure, and a sample path means the function $s\mapsto X(s,\omega )$. The theorem Unbiasedness and L2 consistency of the spatial window average ↗ establishes this integrability and measurability from its stated second-moment hypotheses.