Proof of The Approximate Kalman Filter and Policy for the Controlled N-Agent Dynamics

Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Entries of matrices are handled with the componentwise calculus toolkit: entries of a matrix product are finite sums of products of entries; transposition satisfies $(UV)^{\top}=V^{\top}U^{\top}$; entrywise indefinite Riemann integrals of continuous maps (Riemann integration over compact intervals in the sense of claim 3 of that toolkit) are continuous in the upper limit and additive over subintervals, with the bound that the absolute value of an integral is at most the interval length times the maximum of the absolute integrand; and bilinear forms pass through entrywise integrals, so that, taking one argument over the standard basis vectors, multiplication by a constant vector passes inside an entrywise integral. Finitely many Riemann integrals of continuous functions over a common interval are combined into one by passing to Lebesgue integrals (claim 3 there: for a continuous integrand on a compact interval the two integrals agree) and using the linearity of the integral. For a vector $x$ with $n$ components, $|x|\le\sum_{p=1}^{n}|x^p|$, since $|x|^2=\sum_p(x^p)^2\le\big(\sum_p|x^p|\big)^2$.

Step 1 (Conclusion 1). The components of $t\mapsto(S_t,A_t)$ are continuous by clause 1 of the definition of a mean-field trajectory pair (part of the stationary triple setting), and $S_t\in\Delta^l$ for every $t$.

By the definition of the fluctuation LQG data, the entries of $\mathcal{E}_t$ and $\mathcal{B}_t$ are the values $\partial_i\bar{b}^\gamma(S_t,A_t)$, and each $\partial_i\bar{b}^\gamma$ exists and is of class $C^1$ — in the scalar sense of clause 3 of that definition — on $U\times V$ by part (i) of the regularity of the extended aggregate state drift, hence continuous at every point of $U\times V$ by clause 1 of the $C^k$ definition; moreover $(S_t,A_t)\in\Delta^l\times\mathcal{A}\subseteq U\times V$ for every $t$, since the trajectory pair takes values in $\Delta^l\times\mathcal{A}$ and the extension has $\Delta^l\subset U$ and $\mathcal{A}\subseteq V$; the entries of $\tilde{\mathcal{E}}_t$ are the values $\partial_\gamma\bar{\tilde{b}}^\upsilon(S_t)$, and each $\partial_\gamma\bar{\tilde{b}}^\upsilon$ is a $C^1$ map on $\tilde{U}$ by part (i) of the regularity of the extended aggregate observation drift, hence continuous by clause 1 of the $C^k$ definition. Continuity in $t$ follows by continuity of compositions along the continuous map $t\mapsto(S_t,A_t)$, applied pointwise on $[0,T]$ exactly as in the definition of the fluctuation linear-quadratic cost — the componentwise continuity of clause 1 of the trajectory-pair definition yielding Euclidean continuity of $t\mapsto(S_t,A_t)$ via claim 1 of the agreement of Euclidean and metric continuity and the entry–norm inequalities of claim 1 of the componentwise toolkit, as carried out there.

By the definition of the aggregate fluctuation covariance, each entry of $\Theta^\star_t=\Theta(S_t,A_t)$ is a finite sum of terms $\pm S^\sigma_t\,\beta(\sigma,\gamma,S_t,A_t)$; on $\Delta^l\times\mathcal{A}$ each rate $\beta(\sigma,\gamma,\cdot,\cdot)$ agrees with the restriction of $\bar{\beta}(\sigma,\gamma,\cdot,\cdot)$ by clause 1 of the definition of the transition-rate extension, and $\bar{\beta}(\sigma,\gamma,\cdot,\cdot)$ is of class $C^2$ on $U\times V$ by clause 2 of that definition, hence of class $C^1$ and continuous at every point of $U\times V$ by clauses 2 and 1 of the $C^k$ definition; so each entry of $t\mapsto\Theta^\star_t$ is continuous by composition and sums and products of continuous functions. Likewise $\tilde{b}^\upsilon$ agrees on $\Delta^l$ with the restriction of $\bar{\tilde{b}}^\upsilon$ by part (i) of the observation-drift regularity lemma, and $\bar{\tilde{b}}^\upsilon$ is of class $C^2$ on $\tilde{U}$ by the same part (i), hence of class $C^1$ and continuous by clauses 2 and 1 of the $C^k$ definition; so $t\mapsto\tilde{b}^\upsilon(S_t)$ is continuous, and with it every entry of $t\mapsto\tilde{\Theta}^\star_t$, whose entries are $\mathbf{1}_{\{\upsilon=\upsilon'\}}\tilde{b}^\upsilon(S_t)$ by the LQG data definition. The entries of $Q_t$, $V_t$, $R_t$ are fixed linear combinations of the fluctuation Hessian coefficients $H_{ij}(t)$, each continuous on $[0,T]$ by the statement of the fluctuation LQG cost definition, hence continuous.

For the entry formulas of conclusion 1: by the definition of the fluctuation LQG data, $(H^{SS}_t)_{\gamma\delta}=H_{\gamma\delta}(t)$, $(H^{SA}_t)_{\gamma j}=H_{\gamma,l+j}(t)$, $(H^{AS}_t)_{j\gamma}=H_{l+j,\gamma}(t)$, $(H^{AA}_t)_{jk}=H_{l+j,l+k}(t)$, and $(F^\star)_{\gamma\delta}=F_{\gamma\delta}$; substituting these into the defining matrix formulas for $Q_t$, $V_t$, $R_t$, $\hat{F}$ and using the transpose (which exchanges the two indices) gives exactly the displayed entry formulas.

$R_t$ is symmetric since $R^{ij}_t=\tfrac14\big(H_{l+i,l+j}(t)+H_{l+j,l+i}(t)\big)=R^{ji}_t$. By (H1) its quadratic form is at least $r|a|^2>0$ at every $a\neq0$, so $R_t$ is positive definite and invertible, and the entries of $t\mapsto R_t^{-1}$ are continuous by continuity of the matrix inverse. $\tilde{\Theta}^\star_t$ is symmetric (its off-diagonal entries vanish), and for $y\in\mathbb{R}^{\tilde{l}}$ its quadratic form is $\sum_{\upsilon}\tilde{b}^\upsilon(S_t)(y^\upsilon)^2\ge\tilde{r}|y|^2$ by (H3), so $\tilde{\Theta}^\star_t$ is positive definite and invertible. Let $\Xi_t$ be the diagonal matrix with diagonal entries $1/\tilde{b}^\upsilon(S_t)$, defined since $\tilde{b}^\upsilon(S_t)\ge\tilde{r}>0$; a direct entry computation from the definition of the matrix product gives $\Xi_t\,\tilde{\Theta}^\star_t=\tilde{\Theta}^\star_t\,\Xi_t=I$, so $\Xi_t=(\tilde{\Theta}^\star_t)^{-1}$, whose entries $\mathbf{1}_{\{\upsilon=\upsilon'\}}/\tilde{b}^\upsilon(S_t)$ are continuous in $t$ by continuity of the matrix inverse (or directly, the denominators being continuous and bounded below by $\tilde{r}$). Finally, by (H2) each entry of $t\mapsto Z_t$ is continuous (an entry of a family continuously differentiable in integral form is an integral, with continuous density, of the form appearing in the weighted second-moment evolution lemma, hence continuous in the limit of integration), so the entries of $W_t=Z_t\mathcal{B}_t+\tfrac12V_t$, of $W_t^{\top}$, and of $\mathcal{G}_t=R_t^{-1}W_t^{\top}$ are continuous by sums and products. This proves conclusion 1.

Step 2 (Conclusion 2). $\Xi_t=(\tilde{\Theta}^\star_t)^{-1}$ is symmetric (diagonal). By the transpose identities of the toolkit, $\tilde{D}_t^{\top}=\tilde{\mathcal{E}}_t^{\top}\,\Xi_t^{\top}\,\tilde{\mathcal{E}}_t=\tilde{D}_t$, so $\tilde{D}_t$ is symmetric, and a direct entry computation gives, for $x\in\mathbb{R}^l$,

so $\tilde{D}_t$ is positive semidefinite; its entries are continuous in $t$ as products and sums of continuous entries. Since $(S_t,A_t)\in\Delta^l\times\mathcal{A}$ (Step 1), part 3 of the jump representation lemma applied at $(S_t,A_t)$ shows that $\Theta^\star_t=\Theta(S_t,A_t)$ is positive semidefinite. The global existence and uniqueness theorem for the Kalman covariance Riccati equation, applied on $[0,T]$ with matrix size $l$, coefficient assignments $\mathcal{E}$, $\Theta^\star$, $\tilde{D}$ (continuous, with $\Theta^\star_t$ and $\tilde{D}_t$ positive semidefinite) and initial value $\Pi_0$ (positive semidefinite by (H4)), yields exactly one continuous assignment $\Pi$ satisfying the displayed integral equation, with every $\Pi_t$ symmetric and $0\preceq\Pi_t$ in the semidefinite order, that is, positive semidefinite. The entries of $\tilde{\mathcal{K}}_t=\Pi_t\tilde{\mathcal{E}}_t^{\top}\Xi_t$ and $M_t=\mathcal{E}_t-\mathcal{B}_t\mathcal{G}_t-\tilde{\mathcal{K}}_t\tilde{\mathcal{E}}_t$ are then continuous by sums and products of continuous functions. This proves conclusion 2.

Step 3 (Conclusion 3). Since $M$ has continuous entries, the fundamental-solution theorem on $[0,T]$ provides $\Phi$ with continuous entries satisfying $\Phi(t)=I+\int_0^t M_r\,\Phi(r)\,dr$, invertible at every $t$, with inverse $\Psi$ having continuous entries and satisfying $\Psi(t)=I-\int_0^t\Psi(r)\,M_r\,dr$. Define $p:[0,T]\to\mathbb{R}^l$ by $p_t=\int_0^t\Psi(r)\,\tilde{\mathcal{K}}_r\,\tilde{b}(S_r)\,dr$; its components are continuous by the toolkit (continuity of indefinite Riemann integrals of continuous integrands).

Fix $k\ge1$ and $v\in\{1,\dots,\tilde{l}\}^k$, and write $\mathcal{T}_k$ for the $\sigma$-algebra on $[0,T]\times\mathcal{R}_k$ generated by the relatively open sets, as in the definition of an observation-driven control policy. The coordinate maps $(t,\tau)\mapsto t$ and $(t,\tau)\mapsto\tau_j$ ($j\le k$) are measurable with respect to $\mathcal{T}_k$: the preimage of an open subset $O$ of $\mathbb{R}$ under either map is the intersection of $[0,T]\times\mathcal{R}_k$ with an open subset of $\mathbb{R}^{1+k}$, hence relatively open and in $\mathcal{T}_k$; and since the collection of subsets of $\mathbb{R}$ whose preimages lie in $\mathcal{T}_k$ is a $\sigma$-algebra (preimages commute with complements and countable unions) containing the open sets, it contains every Borel set. The indicator $(t,\tau)\mapsto\mathbf{1}_{\{\tau_j\le t\}}$ is $\mathcal{T}_k$-measurable: the set $\{(t,\tau):\tau_j>t\}$ is the intersection of $[0,T]\times\mathcal{R}_k$ with the open set of points of $\mathbb{R}^{1+k}$ whose $(1+j)$-th coordinate exceeds the first, hence lies in $\mathcal{T}_k$, and so does its complement; every preimage under the indicator is one of $\emptyset$, these two sets, or everything. Every real function of the form $(t,\tau)\mapsto u(t,\tau_j)$ with $u$ sequentially continuous on $[0,T]^2$ in the sense of the composition lemma cited next (the letter $u$ is local to this sentence) — in particular every entry of $\Phi(t)$, $\mathcal{G}_t$, $\Psi(\tau_j)$, $\tilde{\mathcal{K}}_{\tau_j}$, and every component of $A_t$ and $p_t$ — is $\mathcal{T}_k$-measurable by measurability of sequentially continuous functions of measurable Euclidean maps, applied to the measurable pair $(t,\tau)\mapsto(t,\tau_j)$; and finite sums and products of $\mathcal{T}_k$-measurable real functions are $\mathcal{T}_k$-measurable by the same lemma. Each component of $(t,\tau)\mapsto f^N_k(t,\tau,v)$ is such a finite sum of products, hence $\mathcal{T}_k$-measurable, and so is each component of the candidate map $g_{k,v}:(t,\tau)\mapsto A_t-N^{-1/2}\mathcal{G}_t f^N_k(t,\tau,v)$; by its two-case definition, $h^N_k(t,\tau,v)=g_{k,v}(t,\tau)$ if $g_{k,v}(t,\tau)\in\mathcal{A}$ and $h^N_k(t,\tau,v)=A_t$ otherwise.

Measurability of the clamp. By hypothesis (C) the complement of $\mathcal{A}$ is open, so $\mathcal{A}$ is closed and belongs to the Borel $\sigma$-algebra $\mathcal{B}_m$ of $\mathbb{R}^m$ by claim 4 of the Borel measurability lemma on Euclidean space; and since every component of $g_{k,v}$ is $\mathcal{T}_k$-measurable, the map $g_{k,v}$ is measurable from $[0,T]\times\mathcal{R}_k$ with the $\sigma$-algebra $\mathcal{T}_k$ to $\mathbb{R}^m$ with $\mathcal{B}_m$, by the componentwise criterion of claim 2 of the same lemma. Hence the set $g_{k,v}^{-1}(\mathcal{A})=\{(t,\tau):g_{k,v}(t,\tau)\in\mathcal{A}\}$ lies in $\mathcal{T}_k$, and its indicator $\mathbf{1}_{g_{k,v}^{-1}(\mathcal{A})}$ is $\mathcal{T}_k$-measurable. By the two-case definition, at every $(t,\tau)$ and for $j\in\{1,\dots,m\}$,

so each component of $h^N_k(\cdot,\cdot,v)$ is a finite sum of products of $\mathcal{T}_k$-measurable real functions, hence $\mathcal{T}_k$-measurable. For $k=0$, the components of $f^N_0$ and of the candidate map $g_0:t\mapsto A_t-N^{-1/2}\mathcal{G}_t f^N_0(t)$ are continuous in $t$, hence measurable with respect to the $\sigma$-algebra generated by the relatively open subsets of $[0,T]$ by the same preimage argument, and the clamp argument applies verbatim with $g_0$ in place of $g_{k,v}$, giving measurability of the components of $h^N_0$. Thus $h^N$, $f^N$, and $h^{\sharp}$ (whose components are those of $h^N_k$ and $f^N_k$) are observation-driven control policies with horizon $T$, control dimensions $m$, $l$, and $m+l$ respectively, and $\tilde{l}$ channels. Moreover every value of $h^N_k$ (and of $h^N_0$) lies in $\mathcal{A}$: it is either a point of $\mathcal{A}$ by the case condition or the point $A_t$, which lies in $\mathcal{A}$ because the trajectory pair has $A:[0,T]\to\mathcal{A}$; so $h^N$ is $\mathcal{A}$-valued, and every value of $h^{\sharp}_k$ lies in $\mathcal{A}\times\mathbb{R}^l$, so $h^{\sharp}$ is $(\mathcal{A}\times\mathbb{R}^l)$-valued.

For $\beta^{\sharp}$: its members are functions on $\Delta^l\times(\mathcal{A}\times\mathbb{R}^l)$, and $\mathcal{A}\times\mathbb{R}^l$ is a nonempty subset of $\mathbb{R}^{m+l}$, $\mathcal{A}$ being nonempty; clause 1 (bounds) of the definition of a transition-rate family holds since $\beta^{\sharp}(\sigma,\gamma,\Sigma,(a,x))=\beta(\sigma,\gamma,\Sigma,a)\in[0,B]$ for $a\in\mathcal{A}$ by clause 1 for $\beta$; clause 2 (joint sequential continuity) holds because if $(\Sigma_n,c_n)\to(\Sigma,c)$ in $\Delta^l\times(\mathcal{A}\times\mathbb{R}^l)$, then, writing $c_n=(a_n,x_n)$ and $c=(a,x)$, with $a_n,a\in\mathcal{A}$, the Euclidean distance $d(a_n,a)$ is at most $d(c_n,c)$ (a sum of fewer squares under the square root), so $d(a_n,a)\to0$ and $\beta^{\sharp}(\sigma,\gamma,\Sigma_n,c_n)=\beta(\sigma,\gamma,\Sigma_n,a_n)\to\beta(\sigma,\gamma,\Sigma,a)=\beta^{\sharp}(\sigma,\gamma,\Sigma,c)$ by clause 2 for $\beta$. This proves conclusion 3.

Step 4 (Conclusion 4(a)). Fix a solution as in conclusion 4. The projected collection $(\sigma^i,\Upsilon^\upsilon,\alpha,\Omega_0)$ is a solution for $\beta$, $\tilde{\beta}$, the same driving system, and $h^N$: each component of $\alpha$ is a component of $\alpha^{\sharp}$, hence a random variable, and at every $\omega\in\Omega$ and $t\in[0,T]$ the vector $\alpha_t(\omega)$ of the first $m$ components of $\alpha^{\sharp}_t(\omega)\in\mathcal{A}\times\mathbb{R}^l$ lies in $\mathcal{A}$, as the solution concept requires of the control process; the policy $h^N$ is $\mathcal{A}$-valued by conclusion 3, as the solution concept requires of the policy; conditions 1, 4, and 6 of the definition of a solution do not involve the control process and hold as for the given solution; conditions 2 and 3 involve the control only through the values $\beta(\sigma,\gamma,\Sigma_s,\alpha_s)=\beta^{\sharp}(\sigma,\gamma,\Sigma_s,\alpha^{\sharp}_s)$, which are unchanged by the definition of $\beta^{\sharp}$, so the required joint measurability, the consumed clock times, and the counters are identical; and condition 5 holds because at every $\omega\in\Omega_0$ and $t\in[0,T]$, condition 5 for the given solution reads

whose first $m$ components give $\alpha_t=h^N_{\tilde{c}_t}(t,\dots)$ and whose last $l$ components give the displayed identity $\hat{\mathfrak{s}}^N_t=f^N_{\tilde{c}_t}(t,\dots)$. Conversely, any solution for $\beta$, $\tilde{\beta}$, this driving system, and $h^N$ is indistinguishable from the projected solution by the uniqueness part of the existence and uniqueness theorem, both being solutions for the same data.

Combining the identity $\hat{\mathfrak{s}}^N_t=f^N_{\tilde{c}_t}(t,\dots)$ with the two-case definition of $h^N_k$ gives, at every $\omega\in\Omega_0$ and $t\in[0,T]$: the case condition of $h^N_{\tilde{c}_t}$, evaluated along the solution, reads $A_t-N^{-1/2}\mathcal{G}_t\hat{\mathfrak{s}}^N_t\in\mathcal{A}$, which holds exactly when the clamp indicator $\chi_t$ of the statement equals $1$; so $\alpha_t=A_t-N^{-1/2}\mathcal{G}_t\hat{\mathfrak{s}}^N_t$ when $\chi_t=1$ and $\alpha_t=A_t$ when $\chi_t=0$, which is the displayed identity $\alpha_t=\chi_t(A_t-N^{-1/2}\mathcal{G}_t\hat{\mathfrak{s}}^N_t)+(1-\chi_t)A_t$; subtracting $A_t$, multiplying by $N^{1/2}$, and inserting $\mathcal{G}_t=R_t^{-1}W_t^{\top}$ gives the second form, both sides of which vanish when $\chi_t=0$. At $t=0$: by condition 2 the consumed clock times vanish at $0$, and every clock path is a counting path, which vanishes at $0$ by its clause 1; hence every observation counter vanishes at $0$ and $\tilde{c}_0=0$, so $\hat{\mathfrak{s}}^N_0=f^N_0(0)=\Phi(0)\big(-N^{1/2}\,p_0\big)=0$, the integral over the degenerate interval being $0$. This proves (a).

Step 5 (Conclusion 4(b)). Fix $\omega\in\Omega_0$ and abbreviate $\kappa_j=\tilde{\mathcal{K}}_{\tau_j}e_{\upsilon_j}\in\mathbb{R}^l$ (the $\upsilon_j$-th column of $\tilde{\mathcal{K}}_{\tau_j}$) for $j\in\{1,\dots,\tilde{c}_T\}$.

Jump-count identity. By condition 3 of the definition of a solution, the observation total agrees on $[0,T]$ with the restriction of a counting path $c$, and by condition 5 the times $\tau_1<\dots<\tau_{\tilde{c}_T}$ are the jump times of $c$ in $[0,T]$, with $\tilde{c}_t=c(t)$ for $t\in[0,T]$. We claim that for every $t\in[0,T]$ and $j\in\{1,\dots,\tilde{c}_T\}$:

For a natural number $k\ge1$ with $c(u)\ge k$ for some $u\in[0,T]$, let $s_k=\inf\{s\ge0:c(s)\ge k\}\le u$ be the $k$-th jump time of the counting-path definition. Then $c(s)\ge k$ for every $s>s_k$: there is $s'\in(s_k,s]$ with $c(s')\ge k$ by the definition of the greatest lower bound, and $c$ is nondecreasing by clause 2; hence $c(s_k)\ge k$ by right-continuity (clause 3). Also $s_k>0$, since $s_k=0$ would give $c(0)\ge k\ge1$, contradicting $c(0)=0$ (clause 1). For $0\le s<s_k$ we have $c(s)<k$, hence $c(s)\le k-1$, the values being integers (clause 1); so $c(s_k-)\le k-1$, and by the unit-jump clause 4, $c(s_k)\le c(s_k-)+1\le k$. Therefore $c(s_k)=k$ and $c(s_k)>c(s_k-)$, so $s_k$ is a jump time of $c$. The $s_k$ are strictly increasing in $k$ where defined, since $c(s_k)=k$ determines $k$ from $s_k$ and $c$ is a function; $s_k\le T$ whenever $k\le c(T)$; and $s_k>u$ whenever $k>c(u)$, since $s_k\le u$ would give $c(u)\ge c(s_k)=k$ by monotonicity. Conversely, every jump time $v$ of $c$ in $[0,T]$ equals $s_{c(v)}$: with $k=c(v)$ we have $c(v)\ge k$, so $v\ge s_k$; and $c(v)>c(v-)$ forces $c(s)\le c(v-)\le k-1$ for every $s<v$ (monotonicity and integer values), so $s_k\ge v$. Hence the jump times of $c$ in $[0,T]$ are exactly $s_1<s_2<\dots<s_{c(T)}$, so $\tau_j=s_j$ for every $j\in\{1,\dots,\tilde{c}_T\}$, and the claim follows: $\tau_j\le t$ implies $\tilde{c}_t=c(t)\ge c(s_j)=j$ by monotonicity, while $\tilde{c}_t\ge j$ implies $\tau_j=s_j\le t$ by the definition of $s_j$ as a greatest lower bound.

By the jump-count identity, in the identity $\hat{\mathfrak{s}}^N_t=f^N_{\tilde{c}_t}(t,\dots)$ of (a) the sum over $j\le\tilde{c}_t$ may be extended to $j\le\tilde{c}_T$ (the added terms have $\tau_j>t$, so their indicators vanish), which yields the stability identity: for every $t\in[0,T]$,

Two integral identities. First, for every $t\in[0,T]$:

Indeed, componentwise, $(\Phi(t)p_t)^\gamma=\sum_{\delta}\Phi^{\gamma\delta}(t)\,p^\delta_t$, each factor is a constant plus an indefinite Riemann integral of a continuous function, and the product rule for indefinite Riemann integrals gives

summing over $\delta$, combining the finitely many integrals (via claim 3 of the interval toolkit and the linearity of the integral, the combined integrands being continuous), and using $\Phi(r)\Psi(r)=I$ yields the claim. Second, for every $j\in\{1,\dots,\tilde{c}_T\}$ and $t\in[\tau_j,T]$:

since $\Phi(t)-\Phi(\tau_j)=\int_{\tau_j}^{t}M_r\Phi(r)\,dr$ by additivity of the entrywise Riemann integral over subintervals (toolkit), right multiplication by the constant vector $\Psi(\tau_j)\kappa_j$ passes inside the entrywise integral by the bilinear-form identity of the toolkit with the first argument ranging over the standard basis vectors, and $\Phi(\tau_j)\Psi(\tau_j)=I$; for $t=\tau_j$ the identity is trivial, the integral being degenerate.

Assembly. Fix $t\in(0,T]$; for $t=0$ conclusion (b) reads $\hat{\mathfrak{s}}^N_0=0$, which is (a). By claim 3 of the interval toolkit, the first identity's Riemann integral equals, componentwise, the Lebesgue integral over $[0,t]$ of the same (continuous) integrand. For $j$ with $\tau_j<t$, the Riemann integral $\int_{\tau_j}^{t}\phi_j(r)\,dr$ of the continuous map $\phi_j(r)=M_r\,\Phi(r)\,\Psi(\tau_j)\,\kappa_j$ equals, componentwise, the Lebesgue integral over $[\tau_j,t]$ of $\phi_j$ (claim 3 of the toolkit on $[\tau_j,t]$), and this equals the Lebesgue integral over $[0,t]$ of $\mathbf{1}_{\{\tau_j\le r\}}\phi_j(r)$: by claim 2 of the toolkit (zero extension), both are the Lebesgue integral over $\mathbb{R}$ of the same function, namely the restriction of $\phi_j$ to $[\tau_j,t]$ extended by zero. For $j$ with $\tau_j=t$, both $\int_{\tau_j}^{t}\phi_j(r)\,dr$ and the integral of $\mathbf{1}_{\{\tau_j\le r\}}\phi_j(r)$ over $[0,t]$ vanish: the former is degenerate, and the latter has integrand vanishing off the singleton $\{t\}$, so its positive and negative parts have integral $0$ by claim 6 of the toolkit with $D=[0,t)$. Each function $r\mapsto\mathbf{1}_{\{\tau_j\le r\}}$ is measurable on $[0,t]$, the set $[\tau_j,\infty)\cap[0,t]$ lying in the trace Borel $\sigma$-algebra, and finite sums and products of measurable real functions are measurable by measurability of sequentially continuous functions of measurable maps. Therefore, by the stability identity, the two integral identities, and the linearity of the integral over the finitely many terms,

and by the stability identity at $r$ the inner bracket times $\Phi(r)$ is $\hat{\mathfrak{s}}^N_r$, so

Finally, $M_r\hat{\mathfrak{s}}^N_r=\mathcal{E}_r\hat{\mathfrak{s}}^N_r-\mathcal{B}_r\mathcal{G}_r\hat{\mathfrak{s}}^N_r-\tilde{\mathcal{K}}_r\tilde{\mathcal{E}}_r\hat{\mathfrak{s}}^N_r$ by the definition of $M$, an entrywise sum of matrices multiplying a vector splitting into the sum of the matrix-vector products entry by entry, so the integrand equals the one displayed in (b). Its components are measurable on $[0,t]$ (by the stability identity, finite sums of products of continuous functions and the interval indicators above) and bounded there (finitely many terms whose continuous factors attain maximal and minimal values on the compact interval by the extreme value theorem, so their absolute values are bounded), and continuous at every $r$ that is not a jump time, since between consecutive jump times all indicators are constant. This proves (b).

Step 6 (Conclusion 4(c)). Path regularity. Fix $\omega\in\Omega_0$. By the stability identity, each component path is a finite sum of products of continuous functions of $t$ and the functions $t\mapsto\mathbf{1}_{\{\tau_j\le t\}}$, each of which is right-continuous everywhere and continuous except at $\tau_j$; sums and products preserve these properties, so each component path is right-continuous on $[0,T]$ and continuous off $\{\tau_1,\dots,\tau_{\tilde{c}_T}\}$.

Measurability. Applying part (c) of the joint measurability lemma to the given solution for $\beta^{\sharp}$, $\tilde{\beta}$, and $h^{\sharp}$ (a solution of the controlled $N$-agent dynamics whose transition-rate family $\beta^{\sharp}$ has control set $\mathcal{A}\times\mathbb{R}^l$), each map $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,\alpha^{\sharp,j}_t(\omega)$, $j\in\{1,\dots,m+l\}$, is measurable with respect to the product $\sigma$-algebra; taking $j=m+\gamma$ gives the claim for $\mathbf{1}_{\Omega_0}\hat{\mathfrak{s}}^{N,\gamma}$.

Bounds. All entries of $\Phi$, $\Psi$, $\tilde{\mathcal{K}}$, and $\mathcal{G}$ are continuous on $[0,T]$, hence attain maximal and minimal values there by the extreme value theorem, so their absolute values are bounded on $[0,T]$: fix reals $c_\Phi$, $c_\Psi$, $c_K$, $c_G\ge0$ bounding the absolute values of all entries of $\Phi(t)$, $\Psi(t)$, $\tilde{\mathcal{K}}_t$, $\mathcal{G}_t$ on $[0,T]$. By the definition of the aggregate observation drift and the bounds of the observation-rate family, $0\le\tilde{b}^\upsilon(\Sigma)=\sum_{\sigma}\Sigma^\sigma\tilde{\beta}(\sigma,\upsilon,\Sigma)\le\tilde{B}$ for $\Sigma\in\Delta^l$, since the $\Sigma^\sigma$ are nonnegative with sum $1$. Hence each component of $\Psi(r)\tilde{\mathcal{K}}_r\tilde{b}(S_r)$ is at most $l\,\tilde{l}\,c_\Psi c_K\tilde{B}$ in absolute value, each component of $p_t$ is at most $T\,l\,\tilde{l}\,c_\Psi c_K\tilde{B}$ by the integral bounds of the toolkit, and each component of $\Phi(t)p_t$ is at most $l\,c_\Phi\cdot T\,l\,\tilde{l}\,c_\Psi c_K\tilde{B}$. Each $\kappa_j$ is a column of $\tilde{\mathcal{K}}_{\tau_j}$, with components at most $c_K$; so each component of $\Psi(\tau_j)\kappa_j$ is at most $l\,c_\Psi c_K$ and each component of $\Phi(t)\Psi(\tau_j)\kappa_j$ at most $l^2\,c_\Phi c_\Psi c_K$. By the stability identity and $|x|\le\sum_p|x^p|$,

with $C_1=l^3\,\tilde{l}\,c_\Phi c_\Psi c_K\,(1+\tilde{B}T)$, using $\tilde{c}_t\le\tilde{c}_T$. By (a), $N^{1/2}|\alpha_t-A_t|=\chi_t\,|\mathcal{G}_t\hat{\mathfrak{s}}^N_t|\le|\mathcal{G}_t\hat{\mathfrak{s}}^N_t|\le m\,l\,c_G\,|\hat{\mathfrak{s}}^N_t|$, since $\chi_t\in\{0,1\}$ and each of the $m$ components of $\mathcal{G}_t\hat{\mathfrak{s}}^N_t$ is at most $l\,c_G\,|\hat{\mathfrak{s}}^N_t|$ in absolute value. Taking $C^\circ=(1+m\,l\,c_G)\,C_1$, which is determined by the quantities listed in the statement and involves neither $N$ nor the driving system nor the solution, proves both bounds and completes the proof.