Finite modal velocity fields inherit finite second moments

A real Banach space means a real normed linear space complete in its norm metric. This is the codomain condition in [F3] for the phrase “strongly measurable.” Write each $v\in H$ in its $d$ scalar coordinate classes $v^{(r)}\in L^2(D,\lambda ;\mathbb{R})$; the Definition gives the isometric finite direct-sum norm $\mathrm{\parallel }v{\mathrm{\parallel }}_H^2={\sum }_{r=1}^d\mathrm{\parallel }{v}^{(r)}{\mathrm{\parallel }}_2^2$. If $(v_n)$ is Cauchy in this norm, meaning that for each $\epsilon >0$ there is $N$ such that $\mathrm{\parallel }{v}_n-v_q{\mathrm{\parallel }}_H<\epsilon$ for all $n,q\ge N$, then each coordinate sequence is Cauchy because $\mathrm{\parallel }{v}_n^{(r)}-v_q^{(r)}{\mathrm{\parallel }}_2\le \mathrm{\parallel }{v}_n-v_q{\mathrm{\parallel }}_H$. By [F8], each coordinate sequence converges to a class $v^{(r)}\in L^2(D,\lambda ;\mathbb{R})$. The finite tuple $v=(v^{(1)},\dots ,v^{(d)})$ belongs to $H$, and the finite sum of squared coordinate errors tends to zero, so $v_n\to v$ in $H$. Hence $H$ is complete and normed, so it is a real Banach space and [F3] applies.

Put $C_{\varphi }=({\sum }_{j=1}^m\mathrm{\parallel }{\varphi }_j{\mathrm{\parallel }}_H^2{)}^{1\mathrm{/}2}$. For $a,b\in {\mathbb{R}}^m$, the triangle inequality and [F7] give $\mathrm{\parallel }T(a)-T(b){\mathrm{\parallel }}_H\le {\sum }_{j=1}^m\mathrm{\mid }{a}_j-b_j\mathrm{\mid }\mathrm{\parallel }{\varphi }_j{\mathrm{\parallel }}_H\le C_{\varphi }\mathrm{\mid }a-b\mathrm{\mid }\mathrm{.}$ Thus $T$ is continuous and $U=T\circ A$ is Borel measurable. For each $n\ge 1$, truncate every coordinate of $A$ to $[-n,n]$ and round it down to a multiple of $2^{-n}$, obtaining a finite-valued measurable map $A_n:\mathrm{\Omega }\to {\mathbb{R}}^m$ with $A_n\to A$ pointwise. The maps $T\circ A_n$ are measurable simple $H$-valued functions and converge pointwise in norm to $U$ by continuity of $T$. Hence $U$ is strongly measurable by [F3].

For Borel sets $B\subseteq H$, the preimage map preserves complements and countable disjoint unions, and $U^{-1}(H)=\mathrm{\Omega }$. Therefore $\mu (B)=\mathbb{P}(U^{-1}(B))$ is countably additive with $\mu (H)=1$ by [F2]. For any nonnegative Borel function $g:H\to [0,\mathrm{\infty }]$, choose nonnegative simple functions $s_n\uparrow g$ as provided by [F5]. The identity ${\int }_H{s}_{n}d\mu ={\mathbb{E}}_{\mathbb{P}}[s_n(U)]$ follows by writing each simple integral as a finite sum over its level sets and using the definition of $\mu$. Apply [F6] to both sides and pass to the increasing limits to obtain ${\int }_{H}gd\mu ={\mathbb{E}}_{\mathbb{P}}[g(U)]$. Apply this identity to the continuous function $g(v)=\mathrm{\parallel }v{\mathrm{\parallel }}_H^2$. The bound in step 1.2 with $b=0$ gives $\mathrm{\parallel }U{\mathrm{\parallel }}_H^2\le C_{\varphi }^2\mathrm{\mid }A{\mathrm{\mid }}^2$, so ${\mathbb{E}}_{\mathbb{P}}\mathrm{\parallel }U{\mathrm{\parallel }}_H^2\le C_{\varphi }^2{\mathbb{E}}_{\mathbb{P}}\mathrm{\mid }A{\mathrm{\mid }}^2<\mathrm{\infty }$. This proves the stated moment identity and bound, and hence that $\mu$ is a finite-second-moment law of the Definition.