Joint Measurability of the State and Control of the Controlled N-Agent Dynamics

Adopt the setting of the controlled $N$-agent dynamics with $N$ agents, $l$ states, $\tilde{l}$ observation channels, and control dimension $m$, and let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$: a transition-rate family $\beta$ with control set $\mathcal{A}$, an observation-rate family $\tilde{\beta}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$ which is $\mathcal{A}$-valued, and a solution on $[0,T]$ with regular event $\Omega_0$, state processes $\sigma^i_t$, occupation indicators $\eta^{i,\gamma}_t$, empirical state measure $\Sigma_t$ with components $\Sigma^\gamma_t$, control $\alpha_t$, transition counters $N^{i,\sigma\gamma}_t$, and observation counters $\tilde{N}^{i,\upsilon}_t$. Write $\mathbf{1}_{\Omega_0}$ for the function equal to $1$ on $\Omega_0$ and $0$ off $\Omega_0$.

Then each of the following maps on $[0,T]\times\Omega$ is measurable with respect to the product $\sigma$-algebra of the trace Borel $\sigma$-algebra on $[0,T]$ and $\mathcal{F}$, the real line carrying its Borel $\sigma$-algebra:

(a) $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,N^{i,\sigma\gamma}_t(\omega)$ and $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,\tilde{N}^{i,\upsilon}_t(\omega)$, for all $i\in\{1,\dots,N\}$, all ordered pairs $(\sigma,\gamma)$ with $\sigma\neq\gamma$, and all $\upsilon\in\{1,\dots,\tilde{l}\}$;

(b) $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,\eta^{i,\gamma}_t(\omega)$ and $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,\Sigma^\gamma_t(\omega)$, for all $i\in\{1,\dots,N\}$ and $\gamma\in\{1,\dots,l\}$;

(c) $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,\alpha^j_t(\omega)$, for each $j\in\{1,\dots,m\}$.