Proof of Causality of the Mean-Field Flow and Observation-Adaptedness of the Realized Mean-Field Flow

For a measurable space $(\Xi,\mathcal{H})$, call a real-valued map on $\Xi$ $\mathcal{H}$-measurable when it is measurable with respect to $\mathcal{H}$ and $\mathcal{B}(\mathbb{R})$. By Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, a sequentially continuous real-valued function on a nonempty $E\subseteq\mathbb{R}^d$, composed with an $E$-valued map whose components are $\mathcal{H}$-measurable, is $\mathcal{H}$-measurable; we refer to this as (D1). We also use the elementary inequality $|x|\le\sum_{\gamma=1}^{l}|x^\gamma|$ for $x\in\mathbb{R}^l$: by claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, $|x|^2=\sum_\gamma(x^\gamma)^2\le\bigl(\sum_\gamma|x^\gamma|\bigr)^2$, the right side expanding into the left side plus nonnegative cross terms, and the nonnegative square root is nondecreasing by the uniqueness of nonnegative square roots.

Claim 1. Put $x=S(x_0,\xi)$ and $x'=S(x_0,\zeta)$. By claim 2 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls, $x$ is the map furnished by claim 1 of Existence and Uniqueness of the Generalized Mean-Field Trajectory for a Measurable Control for the initial value $x_0$ and the control $u$, and $x'$ the one for $x_0$ and $v$; thus both are continuous, and for every $s\in[0,T]$ and $\gamma$,

$$x^\gamma_s=x^\gamma_0+\int_{[0,s]}\hat{b}^\gamma(x_r,u_r)\,dr,\qquad x'^\gamma_s=x^\gamma_0+\int_{[0,s]}\hat{b}^\gamma(x'_r,v_r)\,dr,$$

the integrands being, for $s\in(0,T]$, bounded and measurable on $[0,s]$: by claims 4 and 6 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data, $|\hat{b}(y,\alpha)|\le K_b$ and $|\hat{b}(y,\alpha)-\hat{b}(y',\alpha')|\le\Lambda_b|y-y'|+2\sqrt{l}\,(l-1)K_1|\alpha-\alpha'|$ for $y,y'\in\mathbb{R}^l$ and $\alpha,\alpha'\in\mathcal{A}$ (the second bound combining, via the triangle inequality of claim 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, the state-Lipschitz bound of claim 6 of the affine-rate lemma with the control-Lipschitz bound of its claim 4 applied at the projected points), so each component $\hat{b}^\gamma$, which satisfies the same bounds by claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, is bounded and sequentially continuous on $\mathbb{R}^l\times\mathcal{A}\subseteq\mathbb{R}^{l+m}$ (both coordinate blocks of a pair converge whenever the pair converges, by the same coordinate bound), and (D1) applies to the continuous map $r\mapsto x_r$ and the measurable map $r\mapsto u_r$ (restricted to $[0,s]$, where measurability with respect to $\mathcal{B}_{[0,s]}$ follows from that with respect to $\mathcal{B}_{[0,T]}$ by intersecting preimages with the Borel set $[0,s]$), and likewise to $x'$ and $v$. If $t=0$ there is nothing to prove, since $x_0=x'_0$. Let $t>0$ and $s\in(0,t]$. We use repeatedly that a bounded $\mathcal{B}_{[0,s]}$-measurable $\varphi:[0,s]\to\mathbb{R}$ with $|\varphi|\le K$ is integrable: $\int_{[0,s]}\varphi^2\,d\lambda_{[0,s]}\le K^2s<\infty$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral and claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, so $|\varphi|$ is integrable by claim 4 of the latter with the constant function $1$, which is what integrability of the measurable $\varphi$ means. In particular the two integrands are integrable, so by linearity, claim 2 of Linearity and Monotonicity of the Lebesgue Integral, $x^\gamma_s-x'^\gamma_s=\int_{[0,s]}g^\gamma(r)\,dr$ with $g^\gamma(r)=\hat{b}^\gamma(x_r,u_r)-\hat{b}^\gamma(x'_r,v_r)$; write $g=(g^1,\dots,g^l)$, whose components are bounded and measurable, and Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval gives that $r\mapsto|g(r)|$ is bounded and measurable on $[0,s]$ and

$$|x_s-x'_s|\le\int_{[0,s]}|g(r)|\,dr .$$

Let $N=\{r\in[0,s]:u(r)\neq v(r)\}$; it belongs to $\mathcal{B}_{[0,s]}$, being the set where the measurable map $r\mapsto|u(r)-v(r)|$ (by (D1), the components of the admissible representatives being measurable by The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval) is positive, and, both $\lambda_{[0,s]}$ and $\lambda_{[0,T]}$ being restrictions of Lebesgue measure $\lambda$ on the real line (claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval), $\lambda_{[0,s]}(N)=\lambda(N)\le\lambda(\{r\in[0,t]:u(r)\neq v(r)\})=0$ by monotonicity of $\lambda$, claim 2 of Basic Properties of a Measure. For $r\in[0,s]\setminus N$ we have $g(r)=\hat{b}(x_r,u_r)-\hat{b}(x'_r,u_r)$, so $|g(r)|\le\Lambda_b|x_r-x'_r|$ by claim 6 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data. The function $w(r)=|x_r-x'_r|$ is continuous on $[0,T]$ (the components of $x-x'$ being continuous and the norm sequentially continuous), hence measurable and Riemann integrable with Lebesgue integral equal to its Riemann integral by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. The product $|g|\mathbf{1}_{[0,s]\setminus N}$ is measurable, since for real $a$ the set where it exceeds $a$ is $([0,s]\setminus N)\cap\{|g|>a\}$ if $a\ge0$ and $[0,s]$ if $a<0$ (claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line), and it is bounded, hence integrable by the remark above; so is $\Lambda_bw$, which is continuous, hence measurable and bounded by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval and Continuous Real-Valued Functions on a Compact Interval are Bounded. Since $|g|$ and $|g|\mathbf{1}_{[0,s]\setminus N}$ agree off the null set $N$, claim 2 of Integrals of Functions Vanishing or Agreeing off a Null Set on a Compact Interval gives $\int_{[0,s]}|g(r)|\,dr=\int_{[0,s]}|g(r)|\mathbf{1}_{[0,s]\setminus N}(r)\,dr$, and the latter integrand is bounded everywhere by $\Lambda_bw(r)$, so by monotonicity

$$w(s)\le\Lambda_b\int_0^s w(r)\,dr\qquad(0<s\le t),$$

while $w(0)=0$. Gronwall's Lemma (Integral Form), applied on $[0,t]$ with $a=0$ and $b=\Lambda_b\ge0$ (nonnegative by its definition in The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data, the quantities $B$, $\Lambda$ and $R$ there being nonnegative), yields $w(s)\le0$ for $s\in[0,t]$; as $w\ge0$, $w=0$ on $[0,t]$, that is, $x_s=x'_s$ for every $s\in[0,t]$ by claim 3 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square.

Claim 2. Fix $x_0$, $t$, $\gamma$ and write $f(\xi)=S^\gamma_t(x_0,\xi)$. Let $a$ be real and $C_a=\{\xi\in\mathcal{U}_{\mathcal{A}}:f(\xi)\le a\}$. We show $C_a$ is sequentially closed in $(\mathcal{U}_{\mathcal{A}},\rho)$. Let $(\xi_n)_{n\in\mathbb{N}}$ be a sequence in $C_a$ converging to $\xi\in\mathcal{U}_{\mathcal{A}}$, so that $\rho(\xi_n,\xi)\to0$. By claim 2 of The Set of Controls with Values in a Compact Convex Set is Weakly Metrizable and Compact, $\xi_n\rightharpoonup\xi$ in the sense of weak convergence. Claim 6 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls, applied with the constant sequence $x^n_0=x_0$, shows that for every $\varepsilon>0$ there is $n_0$ with $|S_t(x_0,\xi_n)-S_t(x_0,\xi)|\le\varepsilon$ for $n\ge n_0$; by claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square the same bound holds for $|f(\xi_n)-f(\xi)|$, so $f(\xi_n)\to f(\xi)$, and $f(\xi)\le a$: otherwise $\varepsilon=f(\xi)-a>0$, and the convergence gives $|f(\xi_n)-f(\xi)|<\varepsilon$, hence $f(\xi_n)>a$, for all large $n$, contradicting $f(\xi_n)\le a$. Hence $\xi\in C_a$. By Sequential Characterization of Closed Subsets of a Metric Space, $C_a$ is closed for the topology of $(\mathcal{U}_{\mathcal{A}},\rho)$, so it belongs to $\mathcal{B}(\mathcal{U}_{\mathcal{A}},\rho)$ by claim 1 of Borel Measurability and Bounded Integration on a Metric Space, and its complement $\{f>a\}$ does as well. Since $a$ was arbitrary, $f$ is measurable with respect to $\mathcal{B}(\mathcal{U}_{\mathcal{A}},\rho)$ and $\mathcal{B}(\mathbb{R})$ by the criterion of claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line.

Claim 3. Fix $t\in[0,T]$ and let $\hat{\alpha}^{(t)}$ be the truncated realized control of claim 4 of Progressive Measurability of the Realized Control with Respect to the Observation Filtration. For every $\omega$, the paths $\hat{\alpha}(\omega)$ and $\hat{\alpha}^{(t)}(\omega)$ are admissible representatives (claim 3 of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set and claim 4 of the progressive measurability lemma) that agree at every point of $[0,t]$, so claim 1 gives $\Phi_t(\omega)=S_t(x_0,\hat{\alpha}(\omega))=S_t(x_0,\hat{\alpha}^{(t)}(\omega))$. The map $\omega\mapsto\hat{\alpha}^{(t)}(\omega)$ is measurable with respect to $\mathcal{G}_t$ and $\mathcal{B}(\mathcal{U}_{\mathcal{A}},\rho)$ by claim 4 of the progressive measurability lemma, and $\xi\mapsto S^\gamma_t(x_0,\xi)$ is measurable with respect to $\mathcal{B}(\mathcal{U}_{\mathcal{A}},\rho)$ and $\mathcal{B}(\mathbb{R})$ by claim 2; their composition $\Phi^\gamma_t$ is $\mathcal{G}_t$-measurable by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. The identities $\Phi_0(\omega)=x_0$ and $|\Phi_t(\omega)-\Phi_r(\omega)|\le K_b|t-r|$, and $\Phi_t(\omega)\in\Delta^l$, are claim 2 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls applied to $S(x_0,\hat{\alpha}(\omega))$. By claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, $|\Phi^\gamma_t(\omega)-\Phi^\gamma_r(\omega)|\le K_b|t-r|$, so each path $t\mapsto\Phi^\gamma_t(\omega)$ is Lipschitz from $[0,T]$ to $\mathbb{R}$, both with the metric of the real line, hence continuous on $[0,T]$ by A Lipschitz Map is Uniformly Continuous and Continuous. A path continuous on $[0,T]$ is right-continuous in the sense of Progressive Measurability: Sections, Right-Continuous Adapted Processes, Arithmetic, and Indefinite Time Integrals: if $(s_j)$ is a sequence in $[s,T]$ converging to $s$, then for every $\varepsilon>0$ the $\delta$ of continuity at $s$ relative to $[0,T]$ shows $|\Phi^\gamma_{s_j}(\omega)-\Phi^\gamma_s(\omega)|<\varepsilon$ for all large $j$. Since $(\Phi^\gamma_t)_{t\in[0,T]}$ is adapted to $(\mathcal{G}_t)_{t\in[0,T]}$ with every path right-continuous, claim 2 of Progressive Measurability: Sections, Right-Continuous Adapted Processes, Arithmetic, and Indefinite Time Integrals makes it progressively measurable with respect to $(\mathcal{G}_t)_{t\in[0,T]}$; and since $\mathcal{G}_t\subseteq\mathcal{F}^{\mathrm{sys}}_t$ for every $t$ (claim 1 of Progressive Measurability of the Realized Control with Respect to the Observation Filtration), every generating rectangle of $\mathcal{B}_{[0,t]}\otimes\mathcal{G}_t$ is one of $\mathcal{B}_{[0,t]}\otimes\mathcal{F}^{\mathrm{sys}}_t$, so the former product $\sigma$-algebra is contained in the latter and the family is progressively measurable with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$ as well.

Claim 4. Fix $t$. The map $Y_t=|\Phi_t-S^*_t|$ is a sequentially continuous function (the Euclidean norm of a difference) of the $\mathcal{G}_t$-measurable maps $\Phi^\gamma_t$ and the constants $S^{*\gamma}_t$, hence $\mathcal{G}_t$-measurable by (D1). For the paths, fix $\omega$ and $r,t\in[0,T]$. By the triangle inequality, claim 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, applied twice, $\bigl|\,|p|-|q|\,\bigr|\le|p-q|$ for $p,q\in\mathbb{R}^l$; with $p=\Phi_t(\omega)-S^*_t$ and $q=\Phi_r(\omega)-S^*_r$ this gives

$$|Y_t(\omega)-Y_r(\omega)|\le|\Phi_t(\omega)-\Phi_r(\omega)|+|S^*_t-S^*_r|\le K_b|t-r|+\sum_{\gamma=1}^{l}|S^{*\gamma}_t-S^{*\gamma}_r| .$$

Given $\varepsilon>0$ and $r\in[0,T]$, continuity of each $S^{*\gamma}$ at $r$ relative to $[0,T]$ provides $\delta_\gamma>0$ with $|S^{*\gamma}_t-S^{*\gamma}_r|<\varepsilon/(2l)$ whenever $t\in[0,T]$ and $|t-r|<\delta_\gamma$; with $\delta=\min(\delta_1,\dots,\delta_l,\varepsilon/(2K_b+2))$ we get $|Y_t(\omega)-Y_r(\omega)|<\varepsilon$ whenever $|t-r|<\delta$. So every path of $Y$ is continuous on $[0,T]$. For the bound, each $S^{*\gamma}$ is bounded on $[0,T]$ by Continuous Real-Valued Functions on a Compact Interval are Bounded, say $|S^{*\gamma}_t|\le c_\gamma$, while $|\Phi_t(\omega)|\le1$ by claim 1 of The Simplex, the Control Set and Their Product are Compact Separable Metric Spaces; hence, by claims 5 and 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square and the inequality $|x|\le\sum_\gamma|x^\gamma|$ noted at the outset, $0\le Y_t(\omega)\le|\Phi_t(\omega)|+|S^*_t|\le1+\sum_\gamma c_\gamma=:K_Y$.

Finally, the family $(Y_t)_{t\in[0,T]}$ consists of random variables on the probability space $(\Omega,\mathcal{G}_T,P)$ (the restriction of $P$ to the sub-$\sigma$-algebra $\mathcal{G}_T$ being a probability measure), each $Y_t$ being $\mathcal{G}_t$-measurable and $\mathcal{G}_t\subseteq\mathcal{G}_T$. Every path is bounded by $K_Y$ and, being continuous on $[0,T]$, is right-continuous at every $t\in[0,T)$ in the sense of The Supremum of a Bounded Right-Continuous Process is a Random Variable (given $\varepsilon>0$, the $\delta$ of continuity at $t$ serves as $\eta$). Applying that lemma on $(\Omega,\mathcal{G}_T,P)$ with $\Omega_0=\Omega$ and $Z_t=Y_t\ge0$ shows that $\overline{Y}=\sup_{t\in D}Y_t$, where $D$ denotes the set of dyadic partition points of $[0,T]$ of that lemma (a symbol used with this meaning only here), is $\mathcal{G}_T$-measurable and equals $\sup_{t\in[0,T]}Y_t(\omega)$ at every $\omega\in\Omega$; in particular the latter supremum exists for every $\omega$. $\blacksquare$