Proof of Joint Measurability of the Tracked Events and of the Tracked Energy Density over the Block Cascade, and Measurability in Time of Their Expectations

Throughout, measurable for a map on $[0,T]\times\Omega$ means measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$ and $\mathcal{B}(\mathbb{R})$, and a subset of $[0,T]\times\Omega$ is called measurable when it belongs to $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$. Fix $k\in\{0,\dots,K-1\}$.

Preliminaries.

Coordinate maps. Throughout, $\Gamma$ denotes a generic Borel set, $\Gamma\in\mathcal{B}(\mathbb{R})$ (the letters $B$ and $S$ being the rate bound and the mean-field state trajectory of the adopted setting). The preimage of $\Gamma$ under $(s,\omega)\mapsto s$ is $(\Gamma\cap[0,T])\times\Omega$, a measurable rectangle in the sense of the product $\sigma$-algebra definition, since $\Gamma\cap[0,T]\in\mathcal{B}_{[0,T]}$ and $\Omega\in\mathcal{F}$; so $(s,\omega)\mapsto s$ is measurable. For a random variable $X$ on $(\Omega,\mathcal{F},P)$ the preimage of $\Gamma$ under $(s,\omega)\mapsto X(\omega)$ is $[0,T]\times X^{-1}(\Gamma)$, again a measurable rectangle; so $(s,\omega)\mapsto X(\omega)$ is measurable. For $D\in\mathcal{F}$ the set $[0,T]\times D$ is a measurable rectangle, and its indicator $(s,\omega)\mapsto\mathbf{1}_{D}(\omega)$ is measurable by claim 1 of the arithmetic of measurable functions. For a function $v:[0,T]\to\mathbb{R}$ continuous on $[0,T]$, the map $s\mapsto v(s)$ is measurable with respect to $\mathcal{B}_{[0,T]}$ by claim 4 of the measurability toolkit, and $(s,\omega)\mapsto v(s)$ is measurable on $[0,T]\times\Omega$, the preimage of $\Gamma$ being $v^{-1}(\Gamma)\times\Omega$ with $v^{-1}(\Gamma)\in\mathcal{B}_{[0,T]}$.

The clock set. By claim 2 of the block cascade lemma, $\sigma^{(k)}$ is a stopping time of the system filtration $(\mathcal{F}^{\mathrm{sys}}_{t})_{t\in[0,T]}$ with values in $[t_{k},T]$; by claim 1 of Stopping Times on a Compact Time Interval: Elementary Operations, the Prior Sigma-Algebra, Dyadic Approximation, Sampling, and Hitting Times, applied to that filtration, $\sigma^{(k)}$ is therefore measurable with respect to $\mathcal{F}^{\mathrm{sys}}_{T}\subseteq\mathcal{F}$ and $\mathcal{B}(\mathbb{R})$, that is, a random variable. By the same claim 2, $G_{k}$ is an event and $G_{0}\supseteq G_{1}\supseteq\dots\supseteq G_{K}$, while $G_{0}=\Omega_{0}$ by the definition of the good sets in the setting of that lemma; so $G_{k}\subseteq\Omega_{0}$. By its claim 1, $t_{0}=0$, $t_{K}=T$ and $t_{k}<t_{k+1}$, so $[t_{k},t_{k+1}]$ is a nondegenerate compact subinterval of $[0,T]$. The map $(s,\omega)\mapsto\sigma^{(k)}(\omega)-s$ is measurable by the coordinate maps and claim 2 of the arithmetic lemma, so

the preimage of the Borel set $(0,\infty)$ under it, is measurable, and

is measurable, a $\sigma$-algebra being closed under intersections. Its indicator, which is the first map of claim 1, is therefore measurable by claim 1 of the arithmetic lemma.

The fluctuation processes on the regular event. By parts (b) and (c) of Joint Measurability of the State and Control of the Controlled N-Agent Dynamics, applied to the transition-rate family $\beta$ with control set $\mathcal{A}$, the family $\tilde\beta$, the horizon $T$, the driving system, the policy $h$ and the solution of the adopted setting, the maps $(s,\omega)\mapsto\mathbf{1}_{\Omega_{0}}(\omega)\Sigma^{\gamma}_{s}(\omega)$ and $(s,\omega)\mapsto\mathbf{1}_{\Omega_{0}}(\omega)\alpha^{j}_{s}(\omega)$ are measurable (that lemma names the trace Borel $\sigma$-algebra on $[0,T]$ through the earlier version of the interval toolkit; it is the same family of sets $S\cap[0,T]$, $S\in\mathcal{B}(\mathbb{R})$). Put, for $(s,\omega)\in[0,T]\times\Omega$,

the second expressions by the definition of the fluctuation processes. Since $s\mapsto S^{\gamma}_{s}$ and $s\mapsto A^{j}_{s}$ are continuous on $[0,T]$, the maps $(s,\omega)\mapsto\mathbf{1}_{\Omega_{0}}(\omega)S^{\gamma}_{s}$ and $(s,\omega)\mapsto\mathbf{1}_{\Omega_{0}}(\omega)A^{j}_{s}$ are measurable by the preliminaries and claim 3 of the arithmetic lemma, and hence each $\bar{\mathfrak{s}}^{\gamma}$ and each $\bar{\mathfrak{a}}^{j}$ is measurable by claim 2 of that lemma. These maps are bounded: $\Sigma_{s}(\omega)$ and $S_{s}$ lie in the probability simplex $\Delta^{l}$ (the empirical state measure by the definition of a solution, $S$ by the adopted setting), and any two points $x,y$ of $\Delta^{l}$ satisfy $|x-y|\le2$, because $|x^{\gamma}-y^{\gamma}|\le1$ gives $(x^{\gamma}-y^{\gamma})^{2}\le|x^{\gamma}-y^{\gamma}|\le x^{\gamma}+y^{\gamma}$ and so $|x-y|^{2}\le2$; hence $|\bar{\mathfrak{s}}_{s}(\omega)|\le2\sqrt{N}$, where $\bar{\mathfrak{s}}_{s}=(\bar{\mathfrak{s}}^{1}_{s},\dots,\bar{\mathfrak{s}}^{l}_{s})$. Likewise $\alpha_{s}(\omega)$ and $A_{s}$ lie in $\mathcal{A}$, the control being $\mathcal{A}$-valued and $A$ taking values in $\mathcal{A}$ by the adopted setting, so $|\alpha_{s}(\omega)-A_{s}|\le|\alpha_{s}(\omega)|+|A_{s}|\le2R$ by the triangle inequality of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and the definition of the control bound, whence $|\bar{\mathfrak{a}}_{s}(\omega)|\le2\sqrt{N}R$ with $\bar{\mathfrak{a}}_{s}=(\bar{\mathfrak{a}}^{1}_{s},\dots,\bar{\mathfrak{a}}^{m}_{s})$.

Claim 1.

The near-field and far-field indicators. The map $(s,\omega)\mapsto|\bar{\mathfrak{a}}_{s}(\omega)|$ is measurable by the composition lemma, the Euclidean norm being a sequentially continuous function on $\mathbb{R}^{m}$ (by claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\bigl||x|-|y|\bigr|\le|x-y|$) and the components $\bar{\mathfrak{a}}^{j}$ being measurable. Hence the set $\mathsf{U}=\{(s,\omega):|\bar{\mathfrak{a}}_{s}(\omega)|\le\sqrt{N}\varrho\}$, the preimage of the Borel set $(-\infty,\sqrt{N}\varrho]$, is measurable. For $\omega\in\mathcal{T}_{k}(s)\subseteq G_{k}\subseteq\Omega_{0}$ one has $\bar{\mathfrak{a}}_{s}(\omega)=\mathfrak{a}_{s}(\omega)$, so

a measurable set, and its indicator, the second map of claim 1, is measurable. The third map is $\mathbf{1}_{\mathcal{T}_{k}(s)}-\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}$ at every point, $\mathcal{T}_{k}(s)$ being the disjoint union of $\mathcal{T}^{\mathrm{nr}}_{k}(s)$ and $\mathcal{T}^{\mathrm{fr}}_{k}(s)$, hence measurable by claim 2 of the arithmetic lemma.

The energy density. Under (H1) and (H2), conclusion (a) of the completion-of-squares theorem gives that every $R_{t}$ is symmetric positive definite, hence invertible, and that all entries of $t\mapsto R_{t}$ and of $t\mapsto\mathsf{M}_{t}=R_{t}^{-1}W_{t}^{\top}$ are continuous on $[0,T]$; write $\mathsf{M}^{i\gamma}_{t}$ ($i\in\{1,\dots,m\}$, $\gamma\in\{1,\dots,l\}$) for the entries of $\mathsf{M}_{t}$. Define on $[0,T]\times\Omega$

Each $\bar{u}^{i}$ is measurable, as a finite sum of products of measurable maps (the entries $\mathsf{M}^{i\gamma}_{s}$ being measurable as continuous functions of $s$, by the preliminaries), by claims 2 and 3 of the arithmetic lemma, and so is $\bar{q}$. For $\omega\in\Omega_{0}$ one has $\bar{\mathfrak{a}}_{s}(\omega)=\mathfrak{a}_{s}(\omega)$ and $\bar{\mathfrak{s}}_{s}(\omega)=\mathfrak{s}_{s}(\omega)$, hence $\bar{u}_{s}(\omega)=u_{s}(\omega)$ and $\bar{q}_{s}(\omega)=u_{s}(\omega)\cdot R_{s}u_{s}(\omega)$ by the entry pairing; since $\mathcal{T}^{\mathrm{nr}}_{k}(s)\subseteq\Omega_{0}$, the fourth map of claim 1 equals $\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}(\omega)\bar{q}_{s}(\omega)$ at every point of $[0,T]\times\Omega$ and is therefore measurable by claim 3 of the arithmetic lemma.

Nonnegativity and boundedness. By (H1), $u_{s}\cdot R_{s}u_{s}\ge r|u_{s}|^{2}\ge0$ at every point, so the fourth map is nonnegative. By the extreme value theorem fix reals $C_{R}\ge0$ and $C_{\mathrm{inv}}\ge0$ bounding in absolute value all entries of $R_{t}$ and of $\mathsf{M}_{t}$ on $[0,T]$. Then at every point, using $|\bar{\mathfrak{a}}^{i}_{s}|\le|\bar{\mathfrak{a}}_{s}|\le2\sqrt{N}R$ and $|\bar{\mathfrak{s}}^{\gamma}_{s}|\le|\bar{\mathfrak{s}}_{s}|\le2\sqrt{N}$, one has $|\bar{u}^{i}_{s}|\le2\sqrt{N}R+2lC_{\mathrm{inv}}\sqrt{N}=:c_{u}$ and hence $|\bar{q}_{s}|\le m^{2}C_{R}c_{u}^{2}=:C_{u}$; since $0\le\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}\le1$, the bound $0\le\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}\,u_{s}\cdot R_{s}u_{s}\le C_{u}$ of claim 1 follows. This proves claim 1.

Claim 2. Let $f$ be any one of the three maps $(s,\omega)\mapsto\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}(\omega)\,u_{s}(\omega)\cdot R_{s}u_{s}(\omega)$, $(s,\omega)\mapsto1-\mathbf{1}_{\mathcal{T}_{k}(s)}(\omega)$ and $(s,\omega)\mapsto\mathbf{1}_{\mathcal{T}^{\mathrm{fr}}_{k}(s)}(\omega)$. Each is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$ by claim 1 (the second by claims 1 and 2 of the arithmetic lemma), and each takes values in $[0,C]$ with $C=\max(C_{u},1)$. The measure spaces $([0,T],\mathcal{B}_{[0,T]},\lambda_{[0,T]})$ and $(\Omega,\mathcal{F},P)$ are finite, hence $\sigma$-finite: the first has total mass $T$ by claim 1 of the interval toolkit, the second total mass $1$. Regarded as a map into $[0,\infty]$, $f$ is measurable in the sense required by Tonelli and Fubini Theorems — that of Lebesgue Integral of a Nonnegative Measurable Function (that theorem names the earlier version of the same definition item; the measurability clause used here is given there by the same words), namely $\{f>a\}\in\mathcal{B}_{[0,T]}\otimes\mathcal{F}$ for every real $a$ — since $\{f>a\}$ is the preimage of the Borel set $(a,\infty)$, $f$ being real-valued; that definition records that for real-valued functions its notion of measurability agrees with measurability with respect to $\mathcal{B}(\mathbb{R})$. By the Tonelli part of Tonelli and Fubini Theorems, the function $F:s\mapsto\int_{\Omega}f(s,\omega)\,dP(\omega)$, with the integral of nonnegative measurable functions, is measurable with respect to $\mathcal{B}_{[0,T]}$ as a $[0,\infty]$-valued function, that is, $\{F>a\}\in\mathcal{B}_{[0,T]}$ for every real $a$; its values lie in $[0,C]$ by the monotonicity of the integral and $P(\Omega)=1$, so $F$ is real-valued and, by the agreement just recalled, measurable with respect to $\mathcal{B}_{[0,T]}$ and $\mathcal{B}(\mathbb{R})$. At each $s$ the section $f(s,\cdot)$ is measurable by the sections part of the Tonelli theorem, nonnegative and bounded, hence a bounded random variable, and $F(s)$ is its expectation $\mathbb{E}[f(s,\cdot)]$, which that definition sets equal to the integral of the nonnegative function $f(s,\cdot)$ with respect to $P$. This gives the measurability on $[0,T]$ of the three functions of claim 2, and their values are real numbers in $[0,C]$. For the restriction to the block $[t_{k},t_{k+1}]$, a nondegenerate compact subinterval of $[0,T]$ as noted in the preliminaries: if $g:[0,T]\to\mathbb{R}$ is measurable with respect to $\mathcal{B}_{[0,T]}$ and $\Gamma\in\mathcal{B}(\mathbb{R})$, then $g^{-1}(\Gamma)=\Gamma'\cap[0,T]$ for some $\Gamma'\in\mathcal{B}(\mathbb{R})$ (the class of sets $\Gamma'\cap[0,T]$ with $\Gamma'$ Borel being exactly $\mathcal{B}_{[0,T]}$), so the preimage of $\Gamma$ under the restriction of $g$ to $[t_{k},t_{k+1}]$ is $g^{-1}(\Gamma)\cap[t_{k},t_{k+1}]=\Gamma'\cap[0,T]\cap[t_{k},t_{k+1}]=\Gamma'\cap[t_{k},t_{k+1}]\in\mathcal{B}_{[t_{k},t_{k+1}]}$; hence the restriction is measurable with respect to $\mathcal{B}_{[t_{k},t_{k+1}]}$.

Finally, the integrals over $[t_{k},t_{k+1}]$ of these three functions — which are respectively the $k$-th summand of the sum in the first display of claim 7, the integral in the third display of claim 1(d), and the integral in the first inequality of claim 1(c) of Ledger Decomposition of the Recentred N-Agent Cost over the Block Cascade and Its Near-Field Filtering Lower Bound — are the integrals of these restrictions with respect to $\lambda_{[t_{k},t_{k+1}]}$, which are Lebesgue integrals over the compact interval of bounded measurable functions; and the measurability of the three restrictions with respect to $\mathcal{B}_{[t_{k},t_{k+1}]}$, just shown, is what hypothesis (MS) of Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Law-Transported Injection Certificates demands of the instance of the present setting formed from each of its solutions, admissible parameter vectors and block indices. This proves claim 2.