Proof of One-Agent-Move Ratio of the Record Density Kernel

Claim 1. The moved initial states are random variables with values in $\{1,\dots,l\}$: for $i\neq i_0$ this is condition 1 of N-Agent Driving System for the original system, and for every state $\delta$ the event where $\varsigma'^{i_0}_0=\delta$ is a finite union of intersections of events of $\sigma(\varsigma^{i_0}_0)$ (it equals the event $\varsigma^{i_0}_0=\delta$ with the event $\varsigma^{i_0}_0=\sigma^*$ removed or, for $\delta=\gamma^*$, adjoined), so $\varsigma'^{i_0}_0$ is measurable and $\sigma(\varsigma'^1_0,\dots,\varsigma'^N_0)\subseteq\sigma(\varsigma^1_0,\dots,\varsigma^N_0)$. Conditions 2 and 3 of N-Agent Driving System concern only the clocks, which are unchanged. For condition 4: by Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras, independence of the finite family consisting of $\sigma(\varsigma'^1_0,\dots,\varsigma'^N_0)$ and the clock $\sigma$-algebras is verified through the product rule for finitely many chosen events, and every event chosen from $\sigma(\varsigma'^1_0,\dots,\varsigma'^N_0)$ is an event of $\sigma(\varsigma^1_0,\dots,\varsigma^N_0)$, for which the product rule holds by condition 4 for the original system. The same inclusion gives $\mathcal{T}'\subseteq\mathcal{T}$: the generators of $\mathcal{T}'$ are events of $\sigma(\varsigma^1_0,\dots,\varsigma^N_0)$ and transition-clock events, all of which lie in $\mathcal{T}$. A solution for $h$ on the moved system exists by claim (ii) of Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics, and Measurable Reconstruction of the Controlled N-Agent Dynamics from Observation Records provides reconstruction data for the moved system and $h$, so the setting of Conditional Density of the Observation Record Given the Initial States and Transition Clocks is fully instantiated for the moved system.

Claim 2. For $(r,\omega)\in G$ with $r=(k,t,v)$, the kernel value $f(r,\omega)$ is the product of the $k$ real factors $N\tilde{b}^{v_j}(\Sigma^r_{t_j-}(\omega))$ with the factor $\exp(-N\int_{[0,T]}\tilde{b}^{\mathrm{tot}}(\Sigma^r_u(\omega))du)$. The integrand is bounded by $\tilde{l}\tilde{B}$, so the integral is a real number and the exponential factor is a strictly positive real, the real exponential function being strictly positive. Each remaining factor is nonnegative, since observation rates are nonnegative by Observation-Rate Family and the coordinates of a member of the probability simplex are nonnegative, so the aggregate observation drift has nonnegative components. A finite product of nonnegative reals and one strictly positive real is strictly positive if and only if every factor is nonzero. This proves claim 2 for $f$; the argument applies verbatim to $f'$ on $G'$.

Claim 3. Let $\omega\in\Omega_G\cap\Omega'_G$ and let $r=(k,t,v)$ satisfy $f(r,\omega)>0$. Since $\mathbf{R}\times\Omega_G\subseteq G$ and $\mathbf{R}\times\Omega'_G\subseteq G'$, both $(r,\omega)\in G$ and $(r,\omega)\in G'$, so $f(r,\omega)$ and $f'(r,\omega)$ are both given by their product-exponential expressions. By claim 2, every factor $\tilde{b}^{v_j}(\Sigma^r_{t_j-}(\omega))$ is strictly positive, so the quotient of the two finite products is the product of the quotients of corresponding factors, the factors $N$ cancelling. For the exponential factors: both integrals are finite; the quotient of values of the exponential function is the exponential of the difference of the arguments, by the functional property of The Real Exponential Function; and the difference of the two integrals is the integral of the difference of the integrands by linearity of the integral, both integrands being bounded and measurable on $[0,T]$ as in the definition of the kernels. Combining gives the asserted identity.

Claim 4. By claim 2 of Conditional Density of the Observation Record Given the Initial States and Transition Clocks, $f$ is measurable with respect to $\mathcal{R}\otimes\mathcal{T}$ and $f'$ with respect to $\mathcal{R}\otimes\mathcal{T}'$; since $\mathcal{T}'\subseteq\mathcal{T}$ by claim 1, and the product $\sigma$-algebra is monotone in its factors (its generators are), both are $\mathcal{R}\otimes\mathcal{T}$-measurable. The set where $f>0$ lies in $\mathcal{R}\otimes\mathcal{T}$. On this set the quotient is the composition of the measurable pair $(f',f)$ with the map $(u,w)\mapsto u/w$, which is sequentially continuous on $[0,\infty)\times(0,\infty)$, so the restriction is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable; the map of claim 4 agrees with this restriction on the set where $f>0$ and with the constant $0$ on its complement, a two-member measurable partition, so it is measurable.