The Control of the Controlled N-Agent Dynamics is Adapted to the Observation Filtration

Adopt the setting of the definition of a \reftext{def:n-agent-controlled-dynamics-2026a}{solution of the controlled $N$-agent dynamics} on $[0,T]$: \reftext{def:natural-numbers-2026a}{natural numbers} $N\ge1$, $l\ge2$, $\tilde{l}\ge1$, $m\ge1$, 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} $h=(h_k)_{k\ge0}$, and a solution with regular event $\Omega_0$, observation processes $\Upsilon^\upsilon$, observation total $\tilde{c}_t$, and control process $\alpha$. Let $(\mathcal{G}_t)_{t\in[0,T]}$ be the observation filtration of the solution definition, and call a random variable $\mathcal{G}_t$-measurable if its preimages of \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel sets} lie in $\mathcal{G}_t$, as in the criterion of the definition of a \reftext{def:measurable-function-2026a}{measurable function}. Then for every $t\in[0,T]$ and every $j\in\{1,\dots,m\}$:

\textbf{1. (Adaptedness.)} There is a $\mathcal{G}_t$-measurable random variable $\alpha'^{\,j}_t$ on $(\Omega,\mathcal{F},P)$ with $\alpha^j_t(\omega)=\alpha'^{\,j}_t(\omega)$ at every $\omega\in\Omega_0$; in particular $\alpha^j_t$ is \reftext{def:almost-surely-2026a}{almost surely} equal to $\alpha'^{\,j}_t$.

\textbf{2. (Square-integrability transfer.)} If $\alpha^j_t$ is \reftext{def:square-integrable-mean-square-2026a}{square-integrable}, then every random variable almost surely equal to $\alpha^j_t$ --- in particular every variable as in conclusion 1 --- is square-integrable with the same mean-square norm.

\textbf{3. (Control fluctuation.)} Consequently, if moreover $(S,A)$ is a \reftext{def:mean-field-trajectory-pair-2026a}{mean-field trajectory pair} for $\beta$ with horizon $T$ --- the trajectory letter $A$ being distinct from the consumed clock times $A^{i,\sigma\gamma}$ of the solution definition --- and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$ is the \reftext{def:n-agent-fluctuation-processes-2026a}{control fluctuation process}, then each component $\mathfrak{a}^j_t$ equals the $\mathcal{G}_t$-measurable random variable $\sqrt{N}(\alpha'^{\,j}_t-A^j_t)$ at every $\omega\in\Omega_0$, hence is almost surely equal to it; and if $\mathfrak{a}^j_t$ is square-integrable, that variable is square-integrable with the same mean-square norm.