Proof of The Record-Frozen Control as a Measurable Map into the Weakly Metrized Control Set, and the Record-Frozen Mean-Field Flow: Joint Measurability, Flow Equation, and Measurability in the Record

Write $\mathcal{G}=\mathcal{B}_{[0,T]}\otimes\mathcal{R}$ for the product $\sigma$-algebra on $[0,T]\times\mathbf{R}$, and $V=\{1,\dots,\tilde{l}\}$ for the set of channels. Components of points of $\mathbb{R}^m$ carry the index $\kappa\in\{1,\dots,m\}$, components of points of $\mathbb{R}^l$ the index $\gamma$.

Devices. Let $(\Xi,\mathcal{H})$ be a measurable space.

(D1) (Arithmetic.) Constants and indicators of members of $\mathcal{H}$ are measurable, and sums, scalar multiples, products, absolute values, maxima and minima of measurable real-valued maps on $\Xi$ are measurable, by claims 1--4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. If $f^{1},\dots,f^{d}:\Xi\to\mathbb{R}$ are measurable, $E\subseteq\mathbb{R}^{d}$ is nonempty with $(f^{1},\dots,f^{d})(\xi)\in E$ for every $\xi$, and $g:E\to\mathbb{R}$ is sequentially continuous on $E$, then $g(f^{1},\dots,f^{d})$ is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable. The Euclidean norm $y\mapsto|y|$ on $\mathbb{R}^{d}$ is sequentially continuous, since $\bigl||y|-|y'|\bigr|\le|y-y'|$ 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 to $y=(y-y')+y'$ and to $y'=(y'-y)+y$; hence the norm of a map with measurable components is measurable. For measurable $f,g$ the sets $\{f\le g\}=(g-f)^{-1}([0,\infty))$ and $\{f<g\}=(g-f)^{-1}((0,\infty))$ lie in $\mathcal{H}$, the two half-lines being Borel sets.

(D2) (Piecewise measurability.) If $(E_{n})_{n}$ is a countable family in $\mathcal{H}$ covering $\Xi$ and $f:\Xi\to\mathbb{R}$ satisfies $E_{n}\cap f^{-1}(A)\in\mathcal{H}$ for every $n$ and every $A\in\mathcal{B}(\mathbb{R})$, then $f$ is measurable, since $f^{-1}(A)=\bigcup_{n}(E_{n}\cap f^{-1}(A))$.

(D3) (Compositions and projections.) A composition of measurable maps is measurable, the preimage of a set under the composition being the preimage under the first map of the preimage under the second. The coordinate projections $(s,r)\mapsto s$ and $(s,r)\mapsto r$ on $[0,T]\times\mathbf{R}$ are measurable with respect to $\mathcal{G}$ and $\mathcal{B}_{[0,T]}$, respectively $\mathcal{R}$, by claim 5 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable; hence a measurable map on $[0,T]$ or on $\mathbf{R}$, regarded as a map on $[0,T]\times\mathbf{R}$ through the projection, is $\mathcal{G}$-measurable. Also $I\times C\in\mathcal{G}$ for $I\in\mathcal{B}_{[0,T]}$ and $C\in\mathcal{R}$ by Product Sigma-Algebra.

(D4) (Nonnegative integrands.) A real-valued map $f\ge0$ on $\Xi$ with $\{f>a\}\in\mathcal{H}$ for every real $a$ is measurable, by the remark on real-valued functions in Lebesgue Integral of a Nonnegative Measurable Function; conversely a measurable real-valued $f\ge0$ is measurable in the sense of that definition.

Step 1: the record $\sigma$-algebra and the event-time coordinates. By The Observation Record Space, $\mathcal{R}$ is the $\sigma$-algebra of the countable disjoint union of the cells, so by claim 4 of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions a set $A\subseteq\mathbf{R}$ belongs to $\mathcal{R}$ if and only if $A\cap C\in\mathcal{F}_{C}$ for every cell $C$, where $\mathcal{F}_{C_\emptyset}=\{\emptyset,C_\emptyset\}$ (claim 3 there) and, for $k\ge1$ and $v\in V^{k}$, $\mathcal{F}_{C_{k,v}}$ is the transport (claim 2 there) along $\mathbf{t}\mapsto(k,\mathbf{t},v)$ of the restriction (claim 1 there) of $\mathcal{B}_{k}$ to the ordered time simplex $D_{k}(T)$, that is,

$$\mathcal{F}_{C_{k,v}}=\bigl\{\{(k,\mathbf{t},v):\mathbf{t}\in E\}:\ E\in\mathcal{B}_{k},\ E\subseteq D_{k}(T)\bigr\}.$$

Every cell belongs to $\mathcal{R}$ by claim 4(a) of that lemma. Consequently a real-valued map $f$ on $\mathbf{R}$ is $\mathcal{R}$-measurable if and only if $f^{-1}(A)\cap C\in\mathcal{F}_{C}$ for every $A\in\mathcal{B}(\mathbb{R})$ and every cell $C$.

For each natural number $i\ge1$ define $\theta_{i}:\mathbf{R}\to\mathbb{R}$ by $\theta_{i}(r)=t_{i}$ if $r=(k,\mathbf{t},v)$ with $k\ge i$, and $\theta_{i}(r)=0$ otherwise. Then $\theta_{i}$ is $\mathcal{R}$-measurable: for $A\in\mathcal{B}(\mathbb{R})$ the set $\theta_{i}^{-1}(A)\cap C$ is $\emptyset$ or $C$ when $C=C_\emptyset$ or $C=C_{k,v}$ with $k<i$ (both lie in $\mathcal{F}_{C}$, with $E=\emptyset$ or $E=D_{k}(T)$ in the second case, $D_{k}(T)\in\mathcal{B}_{k}$ by The Ordered Time Simplex: Borel Measurability and Volume); and for $C=C_{k,v}$ with $k\ge i$ it equals $\{(k,\mathbf{t},v):\mathbf{t}\in E\}$ with $E=D_{k}(T)\cap\pi_{i}^{-1}(A)$, where $\pi_{i}:\mathbb{R}^{k}\to\mathbb{R}$ is the $i$-th coordinate projection, measurable with respect to $\mathcal{B}_{k}$ by claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets; so $E\in\mathcal{B}_{k}$ and $E\subseteq D_{k}(T)$.

Step 2: joint measurability of the record-frozen control. Put $\Theta_{i}(s,r)=\theta_{i}(r)$ and $\mathrm{pr}(s,r)=s$; both are $\mathcal{G}$-measurable by (D3). For $k\ge1$, $v\in V^{k}$ and $j\in\{0,\dots,k\}$ set

$$E_{k,v,j}=\bigl([0,T]\times C_{k,v}\bigr)\cap\{\Theta_{j}\le\mathrm{pr}\}\cap\{\mathrm{pr}<\Theta_{j+1}\},$$

where the factor $\{\Theta_{j}\le\mathrm{pr}\}$ is omitted when $j=0$ and the factor $\{\mathrm{pr}<\Theta_{j+1}\}$ is omitted when $j=k$; and set $E_{\emptyset}=[0,T]\times C_\emptyset$. All these sets lie in $\mathcal{G}$ by (D1) and (D3). Let $r=(k,\mathbf{t},v)\in C_{k,v}$ and $s\in[0,T]$. Since $0<t_{1}<\dots<t_{k}\le T$ by the definition of $D_{k}(T)$, the number $k_{r}(s)$ of indices $i$ with $t_{i}\le s$ equals $j$ exactly when $s<t_{1}$ (case $j=0$), $t_{j}\le s<t_{j+1}$ (case $1\le j\le k-1$), or $t_{k}\le s$ (case $j=k$); and $\Theta_{i}(s,r)=t_{i}$ for $i\le k$. Hence $(s,r)\in E_{k,v,j}$ if and only if $r\in C_{k,v}$ and $k_{r}(s)=j$. As the cells are pairwise disjoint and cover $\mathbf{R}$, the sets $E_{\emptyset}$ and $E_{k,v,j}$ are pairwise disjoint and cover $[0,T]\times\mathbf{R}$; they form a countable family, there being finitely many pairs $(v,j)$ for each $k$.

By The Record-Frozen Control Path and Record-Frozen Policy, for $(s,r)\in E_{k,v,j}$ with $j\ge1$ one has $a^{r}(s)=h_{j}(s,(t_{1},\dots,t_{j}),(v_{1},\dots,v_{j}))=h_{j}(Z_{j}(s,r),v^{(j)})$, where $v^{(j)}=(v_{1},\dots,v_{j})$ and $Z_{j}(s,r)=(s,\Theta_{1}(s,r),\dots,\Theta_{j}(s,r))\in\mathbb{R}^{1+j}$; and for $(s,r)\in E_{\emptyset}$ or $(s,r)\in E_{k,v,0}$ one has $a^{r}(s)=h_{0}(s)=h_{0}(Z_{0}(s,r))$ with $Z_{0}=\mathrm{pr}$.

Fix $j\ge0$, and let $E$ be one of the sets $E_{k,v,j}$ (any $k\ge j$, $k\ge1$, any $v$), or $E=E_{\emptyset}$ when $j=0$; write $v^{(j)}$ for the corresponding mark tuple (empty when $j=0$). The map $Z_{j}$ is measurable with respect to $\mathcal{G}$ and the $\sigma$-algebra $\mathcal{B}_{1+j}$ of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets by claim 2 there; $\mathcal{B}_{1+j}$ contains every open subset of $\mathbb{R}^{1+j}$ by claim 4 there. Put $P_{j}=[0,T]\times R_{j}(T)$, where $R_{j}(T)\subseteq\mathbb{R}^{j}$ is the time-tuple record space of Observation-Driven Control Policy (not to be confused with the bound $R$ or with the observation record space $\mathbf{R}$), and $P_{0}=[0,T]$; let $\mathcal{P}_{j}$ be the $\sigma$-algebra on $P_{j}$ generated by the relatively open sets $U\cap P_{j}$, $U$ open in $\mathbb{R}^{1+j}$. By Observation-Driven Control Policy each component of $p\mapsto h_{j}(p,v^{(j)})$ on $P_{j}$ is measurable with respect to $\mathcal{P}_{j}$. On $E$ the map $Z_{j}$ takes values in $P_{j}$: for $j\ge1$ and $(s,r)\in E$ the tuple $(t_{1},\dots,t_{j})$ satisfies $0\le t_{1}\le\dots\le t_{j}\le T$, so lies in $R_{j}(T)$. Let $\mathcal{W}$ be the family of subsets $W\subseteq P_{j}$ with $E\cap Z_{j}^{-1}(W)\in\mathcal{G}$. Because $Z_{j}$ maps $E$ into $P_{j}$, one has $E\cap Z_{j}^{-1}(P_{j})=E$ and $E\cap Z_{j}^{-1}(P_{j}\setminus W)=E\setminus(E\cap Z_{j}^{-1}(W))$, and preimages commute with countable unions, so $\mathcal{W}$ is a $\sigma$-algebra on $P_{j}$; it contains every relatively open $U\cap P_{j}$, since $E\cap Z_{j}^{-1}(U\cap P_{j})=E\cap Z_{j}^{-1}(U)$ and $U\in\mathcal{B}_{1+j}$. Hence $\mathcal{P}_{j}\subseteq\mathcal{W}$. Now let $A\in\mathcal{B}(\mathbb{R})$ and $\kappa\in\{1,\dots,m\}$. For $(s,r)\in E$ we have $a^{r,\kappa}(s)=h^{\kappa}_{j}(Z_{j}(s,r),v^{(j)})$, so

$$E\cap\{(s,r):a^{r,\kappa}(s)\in A\}=E\cap Z_{j}^{-1}(W),\qquad W=\{p\in P_{j}:h^{\kappa}_{j}(p,v^{(j)})\in A\}\in\mathcal{P}_{j},$$

which lies in $\mathcal{G}$. Device (D2) applied to the countable cover of the previous paragraph shows that $(s,r)\mapsto a^{r,\kappa}(s)$ is $\mathcal{G}$-measurable. This is the first assertion of claim 1.

Step 3: the rest of claim 1. Fix $r\in\mathbf{R}$. By claim 3 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable the section $s\mapsto a^{r,\kappa}(s)$ of the $\mathcal{G}$-measurable map of Step 2 is $\mathcal{B}_{[0,T]}$-measurable, for every $\kappa$. Every value of $a^{r}$ is a value of some member of $h$, hence lies in $\mathcal{A}$ because $h$ is $\mathcal{A}$-valued; thus $|a^{r}(s)|\le R$ for every $s$, $R$ being finite by claim 1 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data. The map $s\mapsto|a^{r}(s)|^{2}$ is measurable by (D1) and bounded by $R^{2}$, so by the monotonicity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral and $\lambda_{[0,T]}([0,T])=T$ (claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval),

$$\int_{[0,T]}|a^{r}|^{2}\,d\lambda_{[0,T]}\le R^{2}T<\infty .$$

Hence $a^{r}$ is square-integrable in the sense of claim 1 of The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval, and since every value lies in $\mathcal{A}$ its class belongs to $\mathcal{U}_{\mathcal{A}}$ by The Set of Controls with Values in a Prescribed Subset of Euclidean Space, with empty exceptional set; being everywhere $\mathcal{A}$-valued, the path is an admissible representative in the sense of claim 2 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls.

Step 4: claim 2. Let $\zeta\in\mathcal{U}_{\mathcal{A}}$ and choose an admissible representative of it, again written $\zeta$ (it exists by claim 2 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls), so that $|\zeta(s)|\le R$ for every $s$; for each $n$ choose a representative of $w_{n}$, again written $w_{n}$, a square-integrable map. Fix $n$ and put

$$f_{n}(s,r)=\bigl(a^{r}(s)-\zeta(s)\bigr)\cdot w_{n}(s)=\sum_{\kappa=1}^{m}\bigl(a^{r,\kappa}(s)-\zeta^{\kappa}(s)\bigr)w^{\kappa}_{n}(s).$$

Each $\zeta^{\kappa}$ and $w^{\kappa}_{n}$ is $\mathcal{B}_{[0,T]}$-measurable, hence $\mathcal{G}$-measurable through the projection (D3), so $f_{n}$ is $\mathcal{G}$-measurable by Step 2 and (D1). By The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ and the triangle inequality (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), $|f_{n}(s,r)|\le|a^{r}(s)-\zeta(s)|\,|w_{n}(s)|\le2R|w_{n}(s)|$. The map $|w_{n}|$ is measurable by (D1) with $\int_{[0,T]}|w_{n}|^{2}\,d\lambda_{[0,T]}<\infty$, so $\int_{[0,T]}|w_{n}|\,d\lambda_{[0,T]}<\infty$ by claim 4 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval.

The measure space $([0,T],\mathcal{B}_{[0,T]},\lambda_{[0,T]})$ is finite (claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval), hence $\sigma$-finite in the sense of Measure, Measure Space, and Probability Measure. The measure space $(\mathbf{R},\mathcal{R},\varrho)$ is $\sigma$-finite as well: $\varrho(C_\emptyset)=1$ and $\varrho(C_{k,v})=\lambda_{k}(D_{k}(T))=T^{k}/k!$ by claims 1, 2, 3 and 4(a) of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions (the measure of a cell under $\varrho$ is its measure under the cell's own measure, which is the unit mass for $C_\emptyset$ and the transported restriction of $\lambda_{k}$ for $C_{k,v}$) and The Ordered Time Simplex: Borel Measurability and Volume, so every cell has finite measure and $\mathbf{R}$ is the union of the countably many cells, which lie in $\mathcal{R}$ (claim 4(a) of Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions). Put $f^{\pm}_{n}=\max(\pm f_{n},0)$, $\mathcal{G}$-measurable by (D1), nonnegative, and measurable in the sense of Lebesgue Integral of a Nonnegative Measurable Function by (D4). The Tonelli statement of Tonelli and Fubini Theorems shows that

$$r\mapsto\int_{[0,T]}f^{\pm}_{n}(s,r)\,d\lambda_{[0,T]}(s)$$

is $\mathcal{R}$-measurable with values in $[0,\infty]$; both integrals are bounded by $2R\int_{[0,T]}|w_{n}|\,d\lambda_{[0,T]}$ by monotonicity (claim 1 of Linearity and Monotonicity of the Lebesgue Integral), hence real, so these maps are $\mathcal{R}$-measurable real-valued maps by (D4). For fixed $r$ the map $s\mapsto f_{n}(s,r)$ is $\mathcal{B}_{[0,T]}$-measurable by claim 3 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable, and integrable, its absolute value being dominated by the integrable map $2R|w_{n}|$, and its integral is the difference of the integrals of $f^{+}_{n}(\cdot,r)$ and $f^{-}_{n}(\cdot,r)$ by that definition; by claim 4 of The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval (the map $a^{r}-\zeta$ being a representative of the difference of the classes by claim 3 there) this integral is $\langle a^{r}-\zeta,w_{n}\rangle_{L^{2}}$. Hence $r\mapsto\langle a^{r}-\zeta,w_{n}\rangle_{L^{2}}$ is $\mathcal{R}$-measurable by (D1), and so is

$$g_{n}(r)=\min\bigl(2^{-n},\ \bigl|\langle a^{r}-\zeta,w_{n}\rangle_{L^{2}}\bigr|\bigr),$$

with values in $[0,1]$. By claim 1 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions the map $r\mapsto\sup_{n}g_{n}(r)$ is $\mathcal{R}$-measurable, and it equals $r\mapsto\rho(a^{r},\zeta)$ by the display defining $\rho$ in The Set of Controls with Values in a Compact Convex Set is Weakly Metrizable and Compact, the supremum existing by claim 1 there. This proves claim 2.

Step 5: claim 3. By claim 3 of The Simplex, the Control Set and Their Product are Compact Separable Metric Spaces the metric space $(\mathcal{U}_{\mathcal{A}},\rho)$ is separable; let $D_{\mathcal{U}}$ be a countable dense subset; it is nonempty, since $\mathcal{U}_{\mathcal{A}}$ is nonempty by claim 1 (applied to the empty record) and a dense subset of a nonempty metric space is nonempty. Then $((\mathcal{U}_{\mathcal{A}},\rho),D_{\mathcal{U}})$ is a separable metric datum in the sense of Borel Sets and Measurable Maps in a Separable Metric Space, and by claim 2 the map $r\mapsto\rho(a^{r},q)$ is $\mathcal{R}$-measurable for every $q\in D_{\mathcal{U}}$; claim 3 of that lemma yields that $r\mapsto a^{r}$ is measurable with respect to $\mathcal{R}$ and $\mathcal{B}(\mathcal{U}_{\mathcal{A}},\rho)$.

Step 6: claim 4. Fix $r\in\mathbf{R}$. By Step 3 the path $a^{r}$ is an admissible representative of the element $a^{r}\in\mathcal{U}_{\mathcal{A}}$, and its components are measurable with respect to $\mathcal{B}_{[0,T]}$. By claim 2 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls, $S(x_0,a^{r})$ is the map $S^{a^{r}}$ 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 $a^{r}$: the unique continuous $x:[0,T]\to\mathbb{R}^l$ with $x^{\gamma}_t=x^{\gamma}_0+\int_{[0,t]}\hat{b}^{\gamma}(x_u,a^{r}(u))\,du$ for all $t$ and $\gamma$ (the integral over $[0,t]$ being read with the convention of claim 4 of the present lemma), and it takes values in $\Delta^l$. Thus $\Phi^{r}=x$, $\Phi^{r}_t\in\Delta^l$, $\Phi^{r}_0=x_0$, and $|\Phi^{r}_t-\Phi^{r}_u|\le K_b|t-u|$ by claim 2 of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls. Since $\Phi^{r}_u\in\Delta^l$, claim 6 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data gives $\hat{b}(\Phi^{r}_u,a^{r}(u))=b(\Phi^{r}_u,a^{r}(u))$ for every $u$, so the displayed identity of claim 4 holds, and $|b(\Phi^{r}_u,a^{r}(u))|\le K_b$ by claim 4 of that lemma.

For the measurability of the integrand fix $\gamma$. By claim 4 of The Transition-Rate Family and Aggregate Drift of Affine-Controlled Data and 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),

$$|b(\Sigma,\alpha)-b(\Sigma',\alpha')|\le|b(\Sigma,\alpha)-b(\Sigma',\alpha)|+|b(\Sigma',\alpha)-b(\Sigma',\alpha')|\le\Lambda_b|\Sigma-\Sigma'|+K_2|\alpha-\alpha'|$$

for all $\Sigma,\Sigma'\in\Delta^l$ and $\alpha,\alpha'\in\mathcal{A}$. Read a point $(\Sigma,\alpha)$ of $\mathbb{R}^l\times\mathbb{R}^m$ as the point $(\Sigma^{1},\dots,\Sigma^{l},\alpha^{1},\dots,\alpha^{m})$ of $\mathbb{R}^{l+m}$, so that $E=\Delta^l\times\mathcal{A}$ is a subset of $\mathbb{R}^{l+m}$. Then $|\Sigma-\Sigma'|$ and $|\alpha-\alpha'|$ are each at most the Euclidean distance of the points $(\Sigma,\alpha)$ and $(\Sigma',\alpha')$ of $\mathbb{R}^{l+m}$, by claims 1 and 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square: that distance is the norm of the difference, whose square is the sum of the squares of the two norms, and both quantities being nonnegative the inequality of squares gives the inequality itself. With claim 4 of that lemma the same bound holds for $|b^{\gamma}(\Sigma,\alpha)-b^{\gamma}(\Sigma',\alpha')|$, so $b^{\gamma}$ is sequentially continuous on $E$ in the sense of Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable: if $(\Sigma_i,\alpha_i)_{i\in\mathbb{N}}$ is a sequence in $E$, $(\Sigma,\alpha)\in E$, and the Euclidean distance from $(\Sigma_i,\alpha_i)$ to $(\Sigma,\alpha)$ tends to $0$, then $|b^{\gamma}(\Sigma_i,\alpha_i)-b^{\gamma}(\Sigma,\alpha)|$ is dominated by $(\Lambda_b+K_2)$ times that distance, hence tends to $0$, so $b^{\gamma}(\Sigma_i,\alpha_i)\to b^{\gamma}(\Sigma,\alpha)$. The maps $u\mapsto\Phi^{r,\gamma'}_u$ ($\gamma'\in\{1,\dots,l\}$) are Lipschitz on $[0,T]$ by the bound $|\Phi^{r}_t-\Phi^{r}_u|\le K_b|t-u|$ just proved together with 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, hence continuous by A Lipschitz Map is Uniformly Continuous and Continuous, hence $\mathcal{B}_{[0,T]}$-measurable by claim 4 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions; and the maps $u\mapsto a^{r,\kappa}(u)$ are $\mathcal{B}_{[0,T]}$-measurable by Step 3. So $u\mapsto b^{\gamma}(\Phi^{r}_u,a^{r}(u))$ is $\mathcal{B}_{[0,T]}$-measurable by (D1), the combined map $u\mapsto(\Phi^{r}_u,a^{r}(u))$ taking values in $E$ since $\Phi^{r}_u\in\Delta^l$ and $a^{r}(u)\in\mathcal{A}$. Its values satisfy $|b^{\gamma}(\Phi^{r}_u,a^{r}(u))|\le|b(\Phi^{r}_u,a^{r}(u))|\le K_b$ 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; being measurable with $\int_{[0,T]}|b^{\gamma}(\Phi^{r}_u,a^{r}(u))|\,du\le K_bT<\infty$ 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, it is integrable, and so is its product with the indicator of $[0,t]$, which is measurable by (D1) and dominated by $|b^{\gamma}(\Phi^{r}_u,a^{r}(u))|$; so the integrals in claim 4 are defined. The final sentence of claim 4 collects the continuity of the components just shown and restates the identity in vector form.

Step 7: claim 5. Fix $t\in[0,T]$ and $\gamma$. By claim 2 of Causality of the Mean-Field Flow and Observation-Adaptedness of the Realized Mean-Field Flow the map $\xi\mapsto S^{\gamma}_t(x_0,\xi)$ on $\mathcal{U}_{\mathcal{A}}$ is measurable with respect to $\mathcal{B}(\mathcal{U}_{\mathcal{A}},\rho)$ and $\mathcal{B}(\mathbb{R})$; composing it with the map of claim 3 (D3) gives that $r\mapsto\Phi^{r,\gamma}_t=S^{\gamma}_t(x_0,a^{r})$ is $\mathcal{R}$-measurable.

For the joint measurability let $q\ge1$ be a natural number and put, for $(t,r)\in[0,T]\times\mathbf{R}$,

$$\phi_{q}(t,r)=\sum_{i=0}^{q-1}\mathbf{1}_{[iT/q,(i+1)T/q)}(t)\,\Phi^{r,\gamma}_{iT/q}+\mathbf{1}_{\{T\}}(t)\,\Phi^{r,\gamma}_{T}.$$

Each summand is the product of the indicator of a set $I\times\mathbf{R}$ with $I\in\mathcal{B}_{[0,T]}$ (a half-open interval or a point, traces of Borel sets) and of an $\mathcal{R}$-measurable map regarded through the projection, so $\phi_{q}$ is $\mathcal{G}$-measurable by (D1) and (D3). For $t\in[iT/q,(i+1)T/q)$ one has $|\phi_{q}(t,r)-\Phi^{r,\gamma}_t|=|\Phi^{r,\gamma}_{iT/q}-\Phi^{r,\gamma}_t|\le|\Phi^{r}_{iT/q}-\Phi^{r}_t|\le K_bT/q$ 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 and claim 4 of the present lemma, and $\phi_{q}(T,r)=\Phi^{r,\gamma}_T$. Hence $(\phi_{q}(t,r))_{q\in\mathbb{N}}$ converges to $\Phi^{r,\gamma}_t$ for every $(t,r)$, and $|\phi_{q}|\le1$ because $|x|\le1$ for $x\in\Delta^l$ (claim 1 of The Simplex, the Control Set and Their Product are Compact Separable Metric Spaces) and a component is bounded by the norm. Claim 2 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions shows that $(t,r)\mapsto\Phi^{r,\gamma}_t$ is $\mathcal{G}$-measurable.

Step 8: claim 6. Each $(t,r)\mapsto S^{*\gamma}_t$ is $\mathcal{G}$-measurable through the projection (D3), so by Step 7 and (D1) (the norm of a map with measurable components) the map $(t,r)\mapsto|\Phi^{r}_t-S^{*}_t|$ is $\mathcal{G}$-measurable; it takes values in $[0,1+K^{*}]$ by the triangle inequality, $|\Phi^{r}_t|\le1$ and $|S^{*}_t|\le K^{*}$. By (D4) it is measurable in the sense of Lebesgue Integral of a Nonnegative Measurable Function, so the Tonelli statement of Tonelli and Fubini Theorems, on the $\sigma$-finite spaces of Step 4, shows that $r\mapsto\int_{[0,T]}|\Phi^{r}_u-S^{*}_u|\,du$ is $\mathcal{R}$-measurable with values in $[0,\infty]$; by monotonicity (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) and $\lambda_{[0,T]}([0,T])=T$ its values lie in $[0,(1+K^{*})T]$, so it is a real-valued $\mathcal{R}$-measurable map by (D4). $\blacksquare$