Statistical states, densities, observables and expectations

Let $({\mathrm{\Gamma }}_N,{\mathcal{F}}_N,m_N)$ be a sigma-finite phase-reference measure space from Classical configuration, cotangent phase and reference measures; the cotangent construction uses the stated Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). A statistical state is a probability measure $P$ on $({\mathrm{\Gamma }}_N,{\mathcal{F}}_N)$, meaning a countably additive measure with $P({\mathrm{\Gamma }}_N)=1$ (Measures on sigma-algebras). A density of $P$ relative to $m_N$ is a measurable $\rho :{\mathrm{\Gamma }}_N\to [0,\mathrm{\infty }]$ such that $P(A)={\int }_A\rho d{m}_N(A\in {\mathcal{F}}_N),{\int }_{{\mathrm{\Gamma }}_N}\rho d{m}_N=1.$ Equivalently, $P=\rho m_N$ in the definition of a measure with density (The measure with density $f$ relative to $\mu$); integration against that density has the meaning supplied by Integrating against a density agrees with integrating the product. A state need not have a density.

An observable is a real-valued measurable function $A:{\mathrm{\Gamma }}_N\to \mathbb{R}$ (Extended-real-valued measurable functions). Its expectation $⟨A{⟩}_P:=\int AdP$ is used only when $A$ is integrable, that is, $\int \mathrm{\mid }A\mathrm{\mid }dP<\mathrm{\infty }$ (Integrable real and complex functions, and their integrals). Its variance is ${\mathrm{Var}}_P(A):=\int (A-⟨A{⟩}_P{)}^{2}dP$ when $A^2$ is integrable. For square-integrable $A,B$, their covariance is ${\mathrm{Cov}}_P(A,B):=\int (A-⟨A{⟩}_P)(B-⟨B{⟩}_P)dP\mathrm{.}$ These expressions are finite under the stated square-integrability hypotheses. Indeed, $P({\mathrm{\Gamma }}_N)=1$ makes $1\in L^2(P)$ with norm $1$. By Cauchy-Schwarz inequality for $L^2$, $\int \mathrm{\mid }A\mathrm{\mid }dP\le (\int A^{2}dP{)}^{1\mathrm{/}2}$, so the mean of a square-integrable observable exists. If $a=⟨A{⟩}_P$, then $\mathrm{\mid }A-a{\mathrm{\mid }}^2\le 2A^2+2a^2$ makes $A-a$ square-integrable; the same argument applies to $B-b$. A second application of that inequality gives $\int \mathrm{\mid }(A-a)(B-b)\mathrm{\mid }dP\le \mathrm{\parallel }A-a{\mathrm{\parallel }}_2\mathrm{\parallel }B-b{\mathrm{\parallel }}_2<\mathrm{\infty }$. Thus both the variance and covariance integrals above are well-defined.

If $P=\rho m_N$ and the integrals are defined, these expectations equal $\int A\rho d{m}_N$ by Integrating against a density agrees with integrating the product.

A macrostate map is a measurable map $M:({\mathrm{\Gamma }}_N,{\mathcal{F}}_N)\to (Y,\mathcal{G})$. Its induced state on $Y$ is the pushforward $M_{\ast }P(B):=P(M^{-1}(B))$ for $B\in \mathcal{G}$. A macrocell associated with a specified measurable cell $B\in \mathcal{G}$ is the measurable phase event $C_B:=M^{-1}(B)$; its reference size is $m_N(C_B)\in [0,\mathrm{\infty }]$. A singleton fibre $M^{-1}(\{y\})$ is used as a macrocell only when $\{y\}\in \mathcal{G}$. For a specified family $\mathcal{C}\subseteq \mathcal{G}$ of cells, an entropy assignment means additional data $S:\mathcal{C}\to [-\mathrm{\infty },\mathrm{\infty }]$; no particular functional of $m_N(C_B)$ is stipulated here. Neither a cell's reference size nor a separately chosen entropy assignment is defined to be the expectation or variance of a microscopic observable.