The Record-Frozen Control as a Measurable Map into the Weakly Metrized Control Set, and the Record-Frozen Mean-Field Flow: Joint Measurability, Flow Equation, and Measurability in the Record

Adopt the setting and notation of the flow stability lemma: an affine-controlled transition-rate family $(\beta_0,\beta_1)$ on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^m$ (nonempty, convex and compact as part of those data) and Lipschitz constant $\Lambda$; its transition-rate family $\beta$ with rate bound $B$, together with the bound $R=\sup_{\alpha\in\mathcal{A}}|\alpha|$, the constants $K_1$ and $\Lambda_b$, the aggregate state drift $b$ and the projected drift $\hat{b}$ of that lemma; a horizon $T>0$; the constants $K_b=2\sqrt{l}\,(l-1)B$ and $K_2=2\sqrt{l}\,(l-1)K_1$ of the flow stability lemma; the probability simplex $\Delta^l$; the notation of the Lebesgue space $L^{2}([0,T];\mathbb{R}^{m})$, including the pairing $\langle\cdot,\cdot\rangle_{L^{2}}$ and the convention of claim 5 there by which an element of that space is denoted by the same symbol as a representative of it; the set $\mathcal{U}_{\mathcal{A}}$ of $\mathcal{A}$-valued controls, its admissible representatives, and the mean-field flow $S(x_0,\xi)$ of claim 2 of the flow stability lemma, with values $S_t(x_0,\xi)\in\Delta^l$; and the metric $\rho$ on $\mathcal{U}_{\mathcal{A}}$ of claim 1 of the weak metrizability and compactness theorem, formed from a fixed sequence $(w_n)_{n\in\mathbb{N}}$ in $L^{2}([0,T];\mathbb{R}^{m})$ whose set of terms is dense there (the index of that sequence, written $r$ in that theorem, is written $n$ here, the letter $r$ being reserved for records), with Borel $\sigma$-algebra $\mathcal{B}(\mathcal{U}_{\mathcal{A}},\rho)$.

Write $|\cdot|$ for the Euclidean norm, $x\cdot y$ for the dot product, $\mathcal{B}(\mathbb{R})$ for the Borel $\sigma$-algebra of the real line, $\mathcal{B}_{[0,T]}$ and $\lambda_{[0,T]}$ for the trace Borel $\sigma$-algebra and restricted Lebesgue measure on $[0,T]$, $\int_{[0,T]}\cdot\,du$ for the Lebesgue integral of a nonnegative function and, for signed integrands, for the Lebesgue integral of an integrable function, both with respect to $\lambda_{[0,T]}$, $\mathcal{B}_k$ and $\lambda_k$ for the $k$-fold Borel $\sigma$-algebra and product Lebesgue measure on $\mathbb{R}^k$, 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 $\mathcal{B}(\mathbb{R})$.

Let $\tilde{l}\ge1$ be a natural number and 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, denoted $\rho(T,\tilde{l})$ in that definition, is written $\varrho$ here to keep it apart from the metric $\rho$. Records are written $r=(k,\mathbf{t},v)$ with $\mathbf{t}=(t_1,\dots,t_k)$ and $v=(v_1,\dots,v_k)$, and $C_\emptyset$, $C_{k,v}$ are the cells. Let $h=(h_k)_{k\ge0}$ be an observation-driven control policy with horizon $T$, control dimension $m$ and $\tilde{l}$ channels that is $\mathcal{A}$-valued, and for $r\in\mathbf{R}$ let $a^{r}:[0,T]\to\mathbb{R}^m$ be its record-frozen control path at $r$, with event count $k_r(s)$ as defined there. Fix $x_0\in\Delta^l$. Then the following hold.

1. (Joint measurability and admissibility.) Each component of the map $(s,r)\mapsto a^{r}(s)$ on $[0,T]\times\mathbf{R}$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$. For every $r\in\mathbf{R}$ the path $a^{r}$ is square-integrable in the sense of claim 1 of The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval, every one of its values lies in $\mathcal{A}$, and $|a^{r}(s)|\le R$ for every $s\in[0,T]$; writing $a^{r}$ also for the element of $L^{2}([0,T];\mathbb{R}^{m})$ it represents, one has $a^{r}\in\mathcal{U}_{\mathcal{A}}$, and the path $a^{r}$ is an admissible representative of it.

2. (Weak distances are measurable in the record.) For every $\zeta\in\mathcal{U}_{\mathcal{A}}$ the map $r\mapsto\rho(a^{r},\zeta)$ on $\mathbf{R}$ is measurable with respect to $\mathcal{R}$.

3. (The record-frozen control is a measurable map into the control set.) The map $r\mapsto a^{r}$ from $\mathbf{R}$ to $\mathcal{U}_{\mathcal{A}}$ is measurable with respect to $\mathcal{R}$ and $\mathcal{B}(\mathcal{U}_{\mathcal{A}},\rho)$.

4. (The record-frozen flow.) For $r\in\mathbf{R}$ and $t\in[0,T]$ put

$$\Phi^{r}_t=S_t(x_0,a^{r})\in\Delta^l ,$$

with components $\Phi^{r,\gamma}_t$ (the record-frozen flow). Then for every $r\in\mathbf{R}$: $\Phi^{r}_0=x_0$; $|\Phi^{r}_t-\Phi^{r}_u|\le K_b|t-u|$ for all $t,u\in[0,T]$; for every $\gamma\in\{1,\dots,l\}$ the map $u\mapsto b^\gamma(\Phi^{r}_u,a^{r}(u))$ on $[0,T]$ is measurable with respect to $\mathcal{B}_{[0,T]}$; $|b(\Phi^{r}_u,a^{r}(u))|\le K_b$ for every $u\in[0,T]$; and

$$\Phi^{r,\gamma}_t=x^{\gamma}_0+\int_{[0,t]}b^\gamma\bigl(\Phi^{r}_u,a^{r}(u)\bigr)\,du\qquad\text{for all }t\in[0,T]\text{ and }\gamma\in\{1,\dots,l\},$$

the integral over $[0,t]$ being that of the integrand multiplied by the indicator of $[0,t]$. Thus $\Phi^{r}:[0,T]\to\Delta^l$ is a map with continuous components satisfying the displayed componentwise identity for every $t$ and $\gamma$; reading integrals of $\mathbb{R}^l$-valued maps componentwise, this is $\Phi^{r}_t=x_0+\int_{[0,t]}b(\Phi^{r}_u,a^{r}(u))\,du$.

5. (Measurability of the flow in the record.) For every $t\in[0,T]$ and every $\gamma\in\{1,\dots,l\}$ the map $r\mapsto\Phi^{r,\gamma}_t$ on $\mathbf{R}$ is measurable with respect to $\mathcal{R}$, and the map $(t,r)\mapsto\Phi^{r,\gamma}_t$ on $[0,T]\times\mathbf{R}$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$.

6. (Integrated deviation from a reference path.) Let $S^{*}:[0,T]\to\mathbb{R}^l$ be a map whose components are measurable with respect to $\mathcal{B}_{[0,T]}$, and let $K^{*}\ge0$ be a real number with $|S^{*}_t|\le K^{*}$ for every $t\in[0,T]$. Then the map $(t,r)\mapsto|\Phi^{r}_t-S^{*}_t|$ on $[0,T]\times\mathbf{R}$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$ with values in $[0,1+K^{*}]$, and the map

$$r\mapsto\int_{[0,T]}|\Phi^{r}_u-S^{*}_u|\,du$$

on $\mathbf{R}$ is measurable with respect to $\mathcal{R}$ with values in $[0,(1+K^{*})T]$.