Finite-second-moment random velocity sensor law

General random-state interface. Let $X$ be a real vector space with a function $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_X:X\to [0,\mathrm{\infty })$ satisfying $\mathrm{\parallel }x{\mathrm{\parallel }}_X=0$ exactly when $x=0$, $\mathrm{\parallel }cx{\mathrm{\parallel }}_X=\mathrm{\mid }c\mathrm{\mid }\mathrm{\parallel }x{\mathrm{\parallel }}_X$ for $c\in \mathbb{R}$, and $\mathrm{\parallel }x+y{\mathrm{\parallel }}_X\le \mathrm{\parallel }x{\mathrm{\parallel }}_X+\mathrm{\parallel }y{\mathrm{\parallel }}_X$. Give $X$ the metric $d_X(x,y):=\mathrm{\parallel }x-y{\mathrm{\parallel }}_X$, and require every Cauchy sequence in this metric to converge in $X$. Also require a subset $D_X\subseteq X$ with an injection into $\mathbb{N}$ such that, for every $x\in X$ and $\epsilon >0$, some $d\in D_X$ satisfies $\mathrm{\parallel }x-d{\mathrm{\parallel }}_X<\epsilon$. These conditions define a complete separable real normed space, that is, a separable real Banach space. Its norm-Borel sigma-algebra is the sigma-algebra generated by the norm-open sets. Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space. For an integrable real-valued measurable function $Z$ on $\mathrm{\Omega }$, write $\mathbb{E}Z:={\int }_{\mathrm{\Omega }}Zd\mathbb{P}$; for nonnegative measurable $Z$, $\mathbb{E}Z$ denotes the extended nonnegative integral. Let $U:\mathrm{\Omega }\to X$ be strongly measurable: measurable finite-valued simple maps converge to $U$ in norm outside a measurable null set. A finite-second-moment state satisfies $\mathbb{E}\mathrm{\parallel }U{\mathrm{\parallel }}_X^2<\mathrm{\infty }$. Finite collections of bounded linear maps $L_j:X\to \mathbb{R}$ are bounded sensors; their operator bounds are $\mathrm{\mid }{L}_j(x)\mathrm{\mid }\le C_j\mathrm{\parallel }x{\mathrm{\parallel }}_X$ with supplied finite $C_j$. When used as a physical model, specify whether randomness describes initial data, forcing, measurement, or an adopted stochastic closure. None of those interpretations follows from deterministic balance equations. This interface asserts no PDE or existence of a measurable solution selection.

Spatial velocity law and integral sensor. Let $d\ge 1$ be an integer and let $(D,\mathcal{D},\lambda )$ be a finite measure space with $0<\lambda (D)<\mathrm{\infty }$, interpreted as a spatial region with volume measure $\lambda$. Let $H:=L^2(D,\lambda ;{\mathbb{R}}^d)$ consist of almost-everywhere equivalence classes of measurable vector fields $v$ with ${\int }_D\mathrm{\mid }v(x){\mathrm{\mid }}^{2}d\lambda (x)<\mathrm{\infty }$, where $\mathrm{\mid }\cdot \mathrm{\mid }$ is the Euclidean norm. Give $H$ the norm $\mathrm{\parallel }v{\mathrm{\parallel }}_H:=({\int }_D\mathrm{\mid }v(x){\mathrm{\mid }}^{2}d\lambda (x){)}^{1\mathrm{/}2}$ and its norm topology; this is the finite direct sum of the scalar $L^2$ norms. Write $\mathcal{B}(H)$ for the sigma-algebra generated by the norm-open subsets of $H$. A finite-second-moment random velocity law is a probability measure $\mu$ on $(H,\mathcal{B}(H))$ such that ${\int }_H\mathrm{\parallel }v{\mathrm{\parallel }}_H^{2}d\mu (v)<\mathrm{\infty }\mathrm{.}$ For a sensor weight $\psi \in H$, define the ideal linear sensor and its centered observable by $S_{\psi }(v):={\int }_{D}v(x)\cdot \psi (x)d\lambda (x),f_{\psi }(v):=S_{\psi }(v)-{\int }_H{S}_{\psi }(w)d\mu (w)\mathrm{.}$ The sensor reads a weighted spatial average; it does not represent a point evaluation. When $v$ is assigned SI velocity units of metres per second and $\lambda$ is volume in ${\mathrm{m}}^d$, assigning $\psi$ units ${\mathrm{m}}^{-d}$ makes $S_{\psi }(v)$ a velocity. Other weight units multiply the sensor units by the corresponding factor. The centered observable is dimensionally the same as the sensor reading.

Strongly measurable Banach-valued function

The space $L^p(\mu )$ as the quotient by null functions

The $L^p$ norm descends to the quotient and makes $L^p$ a normed space for $1\le p\le \mathrm{\infty }$

Minkowski's inequality for integrals, including $p=\mathrm{\infty }$

The Lebesgue integral is linear on $L^1(\mu )$

Probability measures and probability spaces

The L2 velocity sensor is well-defined ↗ supplies representative independence and finiteness of the centering integral.

Teschl, Topics in Real and Functional Analysis, Section 11.6

Tao, An Introduction to Measure Theory, Section 1.7

Terquem, Fluids, Sections 1.1.2 and 6.3.1