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

Adopt the setting of the \reftext{def:n-agent-controlled-dynamics-2026a}{controlled $N$-agent dynamics} with $N$ agents, $l$ states, $\tilde{l}$ observation channels, and control dimension $m$: a \reftext{def:transition-rate-family-2026a}{transition-rate family} $\beta$, an \reftext{def:observation-rate-family-2026a}{observation-rate family} $\tilde{\beta}$, a horizon $T>0$, an \reftext{def:n-agent-driving-system-2026a}{$N$-agent driving system} $(\Omega,\mathcal{F},P)$, an \reftext{def:observation-driven-control-policy-2026a}{observation-driven control policy} $h$, and a \reftext{def:n-agent-controlled-dynamics-2026a}{solution} on $[0,T]$ with regular event $\Omega_0$, state processes $\sigma^i_t$, occupation indicators $\eta^{i,\gamma}_t$, empirical state measure $\Sigma_t$, 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 \reftext{def:measurable-function-2026a}{measurable} with respect to the \reftext{def:product-sigma-algebra-2026a}{product $\sigma$-algebra} of the \reftext{lem:interval-lebesgue-toolkit-2026a}{trace Borel $\sigma$-algebra} on $[0,T]$ and $\mathcal{F}$:

\textbf{(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 $\upsilon\in\{1,\dots,\tilde{l}\}$;

\textbf{(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 $\gamma\in\{1,\dots,l\}$;

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