Periodized empirical field of a finite configuration

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Use the cube ${\mathrm{\Lambda }}_L=[0,L{)}^d$ and the translation $T_x$ defined in Poisson Configuration Reference. For a finite configuration $\omega \subset {\mathrm{\Lambda }}_L$, let ${\omega }^{(L)}=\{q+Lk:q\in \omega ,k\in {\mathbb{Z}}^d\}$ be its periodic continuation. Define its periodized empirical field $R_{L,\omega }:=\frac{1}{L^d}{\int }_{{\mathrm{\Lambda }}_L}{\delta }_{T_x{\omega }^{(L)}}dx\mathrm{.}$ The integral is the push-forward of normalized Lebesgue measure on ${\mathrm{\Lambda }}_L$ under $x\mapsto T_x{\omega }^{(L)}$. For any probability law $C_L$ on finite configurations in ${\mathrm{\Lambda }}_L$, define its empirical-field barycenter as the probability mixture ${\bar{C}}_L(B)=\int C_L(d\omega )R_{L,\omega }(B)$ for every measurable configuration event $B$. By Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, for a nonnegative measurable test $f$, the mixture identity holds with values in $[0,+\mathrm{\infty }]$. For a signed local tame $f$ from Local-Tame Topology, the finite expectation formula ${\bar{C}}_L(f)=\int C_L(d\omega )R_{L,\omega }(f)=\frac{1}{L^d}{\int }_{{\mathrm{\Lambda }}_L}{E}_{C_L}[f(T_x{\omega }^{(L)})]dx$ is asserted when $\int C_L(d\omega )R_{L,\omega }(\mathrm{\mid }f\mathrm{\mid })<\mathrm{\infty }$. In particular $E_{C_L}N<\mathrm{\infty }$ suffices: if a bounded support $A$ is contained in a box of side $a$, then $N_A(T_x{\omega }^{(L)})\le (\lceil a\mathrm{/}L\rceil +2{)}^{d}N(\omega )$ uniformly in $x$. The growth bound $\mathrm{\mid }f\mathrm{\mid }\le b(1+N_A)$ therefore gives absolute integrability. For stationary laws write $\rho (P)=E_P{N}_{[0,1{)}^d}\in [0,+\mathrm{\infty }]$, with finite intensity meaning $\rho (P)<\mathrm{\infty }$; this agrees with the finite-intensity convention in Superstable regular pair interactions and configurational ensemble notation. This is a spatial average over the finite box, not the raw finite-volume law viewed at a fixed origin.