The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set

Data. Adopt the setting of the realized-control lemma, with its transition-rate family specialised as follows: natural numbers $N\ge1$, $l\ge2$, $\tilde{l}\ge1$ and $m\ge1$; a real number $\Lambda\ge0$ and an affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^{m}$ and Lipschitz constant $\Lambda$, the set $\mathcal{A}$ being nonempty, convex and compact for the topology of the Euclidean distance; the number $R=\sup_{\alpha\in\mathcal{A}}|\alpha|$ of the affine rate family lemma, finite by claim 1 there, so that $|a|\le R$ for every $a\in\mathcal{A}$, $|\cdot|$ being the Euclidean norm; the transition-rate family $\beta$ of $(\beta_{0},\beta_{1})$, which is a transition-rate family on $l$ states with control set $\mathcal{A}$ by claim 2 of that lemma and is the transition-rate family of the adopted setting, the number $R$ serving as the bound on $\mathcal{A}$ used there, and the point $a_{0}\in\mathcal{A}$ fixed there (which fixes the value of the realized control off the regular event); an observation-rate family $\tilde{\beta}$ on $l$ states with $\tilde{l}$ channels; a real number $T>0$; an $N$-agent driving system $(\Omega,\mathcal{F},P)$; an $\mathcal{A}$-valued observation-driven control policy $h=(h_{k})_{k\ge0}$ with horizon $T$, control dimension $m$ and $\tilde{l}$ channels; and a solution of the controlled $N$-agent dynamics on $[0,T]$ for these data, with regular event $\Omega_{0}$, control $\alpha$, observation total $\tilde{c}$, observation-event count $K_{t}=\tilde{c}_{t}$, observation event times and channels as in condition 5 of that definition, and observation record $W$. Let $(\mathbf{R},\mathcal{R})=(\mathbf{R}(T,\tilde{l}),\mathcal{R}(T,\tilde{l}))$ be the observation record space with horizon $T$ and $\tilde{l}$ channels together with its record $\sigma$-algebra, its reference measure being written $\varrho$ and its records $r=(k,\mathbf{t},v)$; let $r_{\emptyset}$ be the empty record. The random variables $\tau_{j}$ and $\upsilon_{j}$ ($j\ge1$) are those furnished by claim 1 of the realized-control lemma, so that $\tilde{c}_{t}(\omega)\ge j$ if and only if $\tau_{j}(\omega)\le t$, for all $\omega\in\Omega$, $j\ge1$ and $t\in[0,T]$, and at every $\omega\in\Omega_{0}$, for $1\le j\le K_{T}(\omega)$, $\tau_{j}(\omega)$ is the $j$-th jump time of the observation total and $\upsilon_{j}(\omega)$ the channel attached to it in condition 5 (that these are the event times of condition 5 listed in increasing order, with their channels, is established in the proof of claim 1 below); the realized control $\hat{\alpha}:[0,T]\times\Omega\to\mathbb{R}^{m}$ is that of claim 2 of the realized-control lemma, formed from this family. For $r\in\mathbf{R}$ let $a^{r}:[0,T]\to\mathbb{R}^{m}$ be the record-frozen control path of $h$ at $r$, with event count $k_{r}(s)$ as defined there; by claim 1 of the record-frozen control lemma (whose setting is instantiated by the present data on choosing any initial state $x_{0}$ in the probability simplex $\Delta^{l}$, which is nonempty as it contains the first standard basis vector, the metric $\rho$ and its dense sequence in the Lebesgue space $L^{2}([0,T];\mathbb{R}^{m})$ being those already fixed in the adopted setting; neither $x_{0}$ nor $\rho$ enters that claim), each component of $(s,r)\mapsto a^{r}(s)$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$ and $|a^{r}(s)|\le R$ for all $s$ and $r$.

Comparison data. Let $S^{*}:[0,T]\to\mathbb{R}^{l}$, $t\mapsto S^{*}_{t}$, be a map each of whose components is continuous on $[0,T]$, the interval and the real line carrying the metric of the real line, and let $K^{*}\ge0$ be a real number with $|S^{*}_{t}|\le K^{*}$ for every $t\in[0,T]$ (the symbols $S^{*}$ and $K^{*}$ are those of claim 6 of the record-frozen control lemma, the present continuity assumption being stronger than the measurability assumed there; the mean-field flow $S(\cdot,\cdot)$ of the setting of that lemma is not used here). Let $A:[0,T]\to\mathcal{A}$, $t\mapsto A_{t}$, be a map each of whose components is measurable with respect to $\mathcal{B}_{[0,T]}$.

Paths. Let $E$ be a nonempty finite set of points of $\mathbb{R}^{l}$, let $\mathsf{Path}(E,T)$ be the space of piecewise constant paths in $E$ with horizon $T$, with its $\sigma$-algebra $\mathcal{C}_{T}$ generated by the sets $\{p:p(u)=y\}$ ($u\in[0,T]$, $y\in E$), and put $K_{E}=\max_{y\in E}|y|$. Let $\mathcal{D}$ be the set of dyadic partition points of $[0,T]$, that is, the set of all numbers $iT2^{-n}$ with $n$ a natural number or zero and $i\in\{0,1,\dots,2^{n}\}$.

Conventions. $\mathcal{B}_{[0,T]}$ and $\lambda_{[0,T]}$ are the trace Borel $\sigma$-algebra and restricted Lebesgue measure on $[0,T]$; $\int_{[0,T]}\cdot\,du$ is the Lebesgue integral of a nonnegative measurable function with respect to $\lambda_{[0,T]}$; $\otimes$ denotes the product $\sigma$-algebra; and a real-valued map on a measurable space is called measurable when it is measurable with respect to the named $\sigma$-algebra and the Borel $\sigma$-algebra of the real line. Notational cautions: the bound $R$ is unrelated to the record space $\mathbf{R}$ and to the record spaces $R_{k}(T)$ of the policy definition; the reference measure of the record space is written $\varrho$ to keep it apart from the metric $\rho$ on the control set fixed in the adopted setting; the sans-serif letters $\mathsf{s}$, $\mathsf{d}$ and $\mathsf{E}$ are unrelated to the time variable $s$, to the dimension $d$ of other lemmas and to the finite set $E$; the dyadic set $\mathcal{D}$ is unrelated to the ordered time simplices $D_{k}(T)$ of the record space; and the index of the dense sequence in the adopted setting of the realized-control lemma, written $r$ there, plays no role here, the letter $r$ being reserved for records.

Then the following hold.

1. (The realized control is the record-frozen control at the record.) For every $\omega\in\Omega_{0}$ and every $t\in[0,T]$,

2. (The control discrepancy of a record.) For every $r\in\mathbf{R}$ the map $u\mapsto|a^{r}(u)-A_{u}|$ on $[0,T]$ is $\mathcal{B}_{[0,T]}$-measurable with values in $[0,2R]$, so that the control discrepancy

is a real number with $0\le\mathsf{d}(r)\le2RT$. The map $(u,r)\mapsto|a^{r}(u)-A_{u}|$ on $[0,T]\times\mathbf{R}$ is $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$-measurable, and the map $r\mapsto\mathsf{d}(r)$ on $\mathbf{R}$ is $\mathcal{R}$-measurable. Moreover, for every $\omega\in\Omega_{0}$,

3. (The deviation of a path from $S^{*}$.) For every $p\in\mathsf{Path}(E,T)$ the set $\{|p(u)-S^{*}_{u}|:u\in[0,T]\}$ is bounded above by $K_{E}+K^{*}$, and its least upper bound, the path deviation

satisfies $\mathsf{s}(p)=\sup_{q\in\mathcal{D}}|p(q)-S^{*}_{q}|$ and $0\le\mathsf{s}(p)\le K_{E}+K^{*}$. The map $p\mapsto\mathsf{s}(p)$ on $\mathsf{Path}(E,T)$ is $\mathcal{C}_{T}$-measurable.

4. (The closeness set.) For all real numbers $\varepsilon\ge0$ and $\varepsilon'\ge0$ the closeness set

belongs to $\mathcal{C}_{T}\otimes\mathcal{R}$.