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

Setting. Adopt the setting, notation, parameters and standing hypotheses of Ledger Decomposition of the Recentred N-Agent Cost over the Block Cascade and Its Near-Field Filtering Lower Bound, formed for one and the same data as there. In particular: the natural numbers $N\ge1$, $l\ge2$, $m\ge1$ and $\tilde{l}\ge1$ (the number of observation channels); the affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states with compact convex control set $\mathcal{A}\subseteq\mathbb{R}^{m}$ and control bound $R=\sup_{a\in\mathcal{A}}|a|$, and its transition-rate family $\beta$; the observation-rate family $\tilde\beta$; the horizon $T>0$; the $N$-agent driving system $(\Omega,\mathcal{F},P)$ with expectation $\mathbb{E}$; the $\mathcal{A}$-valued observation-driven control policy $h$ and the solution of the controlled $N$-agent dynamics on $[0,T]$ for it, with regular event $\Omega_{0}$, empirical state measure $\Sigma$ with components $\Sigma^{\gamma}$ and control $\alpha$ with components $\alpha^{j}$; the stationary mean-field triple $(S,A,P)$, whose first two components form a mean-field trajectory pair and hence have continuous components on $[0,T]$ by that definition; the fluctuation processes $\mathfrak{s}_{t}=\sqrt{N}(\Sigma_{t}-S_{t})$ and $\mathfrak{a}_{t}=\sqrt{N}(\alpha_{t}-A_{t})$, maps on $\Omega$ with values in $\mathbb{R}^{l}$ and $\mathbb{R}^{m}$; the block data $K$ and $t_{0}<t_{1}<\dots<t_{K}$, the anchored clocks $\sigma^{(k)}$ ($k\in\{0,\dots,K-1\}$) and the good sets $G_{k}$ ($k\in\{0,\dots,K\}$) of the block cascade lemma; the matrices $R_{t}$ and $W_{t}=Z_{t}\mathsf{B}_{t}+\tfrac12V_{t}$ of the completion-of-squares theorem, its hypothesis (H1) with the constant $r=c_{J}$, its hypothesis (H2) with the Riccati family $Z=(Z_{t})_{t\in[0,T]}$ (assumed in the ledger lemma), and its entry pairing $x\cdot My=\sum_{p,q}M^{pq}x^{p}y^{q}$; the process $u_{t}=\mathfrak{a}_{t}+R_{t}^{-1}W_{t}^{\top}\mathfrak{s}_{t}$; the near-field radius $\varrho>0$; and, for $k\in\{0,\dots,K-1\}$ and $s\in[0,T]$, the tracked events

of its derived-quantities paragraph. Write $|\cdot|$ for the Euclidean norm, $\mathbf{1}_{D}$ for the indicator of a set $D$, $\mathcal{B}(\mathbb{R})$ for the Borel $\sigma$-algebra of the real line, $\mathcal{B}_{[a,b]}$ for the trace Borel $\sigma$-algebra on a compact interval $[a,b]$ (the family of the sets $\Gamma\cap[a,b]$ with $\Gamma\in\mathcal{B}(\mathbb{R})$) and $\lambda_{[a,b]}$ for the restricted Lebesgue measure on it, and $\otimes$ for the product $\sigma$-algebra. A real-valued map on a measurable space is called measurable when it is measurable with respect to the named $\sigma$-algebra and $\mathcal{B}(\mathbb{R})$. Notational cautions: $R$ is the control bound and $R_{t}$ a coefficient matrix; $P$ is the probability of the driving system and $P_{t}$ the co-state; $K$ is the number of blocks; the anchored clocks $\sigma^{(k)}$ carry a parenthesised block index and are unrelated to the state processes $\sigma^{i}$ of the solution; the letters $\mathcal{E}$ (the control energy), $Q$ (the noise majorant) and $\mathcal{N}$ (the noise event) of the adopted setting are not used here; and $\Gamma$ denotes a generic Borel subset of the real line.

Then the following hold.

1. (Joint measurability of the tracked indicators and of the tracked energy density.) For every $k\in\{0,\dots,K-1\}$ the four maps on $[0,T]\times\Omega$

are measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$, and the fourth is nonnegative and bounded: there is a real $C_{u}\ge0$ (depending on $N$) with $0\le\mathbf{1}_{\mathcal{T}^{\mathrm{nr}}_{k}(s)}(\omega)\,u_{s}(\omega)\cdot R_{s}u_{s}(\omega)\le C_{u}$ for all $(s,\omega)\in[0,T]\times\Omega$.

2. (Measurability in time of the tracked densities.) For every $k\in\{0,\dots,K-1\}$ the three functions

are real-valued and measurable on $[0,T]$ with respect to $\mathcal{B}_{[0,T]}$, and their restrictions to the block $[t_{k},t_{k+1}]$ are measurable with respect to $\mathcal{B}_{[t_{k},t_{k+1}]}$. In particular the integrals of these three functions over $[t_{k},t_{k+1}]$ — 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 Lebesgue integrals over the compact interval of bounded measurable functions; and the measurability of the three restrictions with respect to $\mathcal{B}_{[t_{k},t_{k+1}]}$ 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, for each of its solutions, admissible parameter vectors and block indices, of the instance of the present setting formed from them.