Proof of Stopped Weighted Second-Moment Evolution of the State Fluctuation Process

Write $\mathbf{1}=\mathbf{1}_{\Omega_0}$ (equal to $1$ on the regular event $\Omega_0$ and $0$ off it). Since $\Omega_0$ has probability $1$, expectations are unchanged when integrands are modified off $\Omega_0$, and we use this silently. Write $I=(I_s)_{s\in[0,T]}$ for the pre-stopping-time indicator of $\tau$, so that $I_s=\mathbf{1}_{\{s<\tau\}}$ in the notation of the statement; by claim 1 of that lemma $I$ is progressively measurable with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$, so each $I_s$ is $\mathcal{F}^{\mathrm{sys}}_s$-measurable and $(s,\omega)\mapsto I_s(\omega)$ is product-measurable by claim 1 of the progressive measurability toolkit. For a family $X$ indexed by $[0,T]$ we write $\hat{X}_t=X_{\min(t,\tau)}$ for the sampled function at $\min(t,\tau)$ — a stopping time by claim 1 of the stopping-time toolkit — so $\hat{\mathfrak{s}}^\gamma_t=\mathfrak{s}^\gamma_{\min(t,\tau)}$. We use throughout the stopped covariation lemma, whose setting is contained in the present one, with its constant $K_M=1+2(l-1)BT$. The event $\Omega\setminus\Omega_0$ has probability zero, and by the solution definition the system filtration contains every probability-zero event of $\mathcal{F}$; hence $\Omega_0\in\mathcal{F}^{\mathrm{sys}}_t$ for every $t$ and $\mathbf{1}$ is $\mathcal{F}^{\mathrm{sys}}_0$-measurable. Finally, a measurable real-valued function bounded in absolute value by a real $c\ge0$ is integrable on a finite measure space: its absolute value has integral at most $c$ times the total mass, by monotonicity of the nonnegative integral against the constant $c$, whose integral is $c$ times the total mass directly from the definition of the nonnegative integral (a simple function). We use this bounded-integrability remark, without further comment, at every application below of the integration by parts lemma and of the Fubini theorem: every integrand concerned is measurable and bounded on $[0,t]$ with the restricted Lebesgue measure, on $[0,T]\times\Omega$ with the finite product measure, or on $(\Omega,\mathcal{F},P)$. All uses of the Tonelli and Fubini theorems are on the product of $[0,T]$ (trace Borel $\sigma$-algebra, restricted Lebesgue measure, total mass $T$ by the toolkit) with the probability space $(\Omega,\mathcal{F},P)$, both finite, hence $\sigma$-finite.

Step 0 (measurability and bounds). Every $\Sigma_t$ lies in the probability simplex (each agent occupies exactly one state, by the derived notation of the solution definition), so $|\mathfrak{s}_t|\le2\sqrt{N}$ everywhere, any two points of the simplex having Euclidean norm at most $1$. By the joint measurability lemma and measurability of sequentially continuous functions of measurable maps, the maps $\mathbf{1}\mathfrak{s}^\gamma_t$, $\mathbf{1}b^\gamma(\Sigma_t,\alpha_t)$, and $\mathbf{1}\Theta^{\gamma\delta}(\Sigma_t,\alpha_t)$ are product-measurable: for the latter two, replace $(\Sigma_t,\alpha_t)$ off $\Omega_0$ by a fixed point of $\Delta^l\times\mathcal{A}$ (which is nonempty) and compose the componentwise-measurable modified map with the sequentially continuous functions $b^\gamma$ and $\Theta^{\gamma\delta}$, whose sequential continuity follows from the joint continuity clause of the transition-rate family and continuity of the coordinate factors in their defining formulas; for $\mathbf{1}\mathfrak{s}^\gamma_t$, subtract the product-measurable $(t,\omega)\mapsto\mathbf{1}S^\gamma_t$ (continuous in $t$, via the composition lemma applied to $(t,\omega)\mapsto t$). On $\Omega_0$, $|b^\gamma|\le2(l-1)B$ and $|\Theta^{\gamma\delta}|\le2(l-1)B$ by part (a) of the martingale decomposition theorem, so $|g^\gamma_s|\le4\sqrt{N}(l-1)B$ there; moreover, by the final sentence of part (a), $|\mathbf{1}b^\gamma(\Sigma_s,\alpha_s)|\le2(l-1)B$ at every point of $\Omega$, and $|b^\gamma(S_s,A_s)|\le2(l-1)B$ for every $s$ by the formula of the aggregate state drift with $0\le\beta\le B$ (transition-rate family) and $S_s$ in the simplex, so $|\mathbf{1}g^\gamma_s|\le4\sqrt{N}(l-1)B$ at every point of $\Omega$, and $(s,\omega)\mapsto\mathbf{1}g^\delta_s=\sqrt{N}\,\big(\mathbf{1}b^\delta(\Sigma_s,\alpha_s)-\mathbf{1}\,b^\delta(S_s,A_s)\big)$ is product-measurable as well, the deterministic $s\mapsto b^\delta(S_s,A_s)$ being continuous by condition 2 of the mean-field trajectory pair, hence product-measurable via composition with $(s,\omega)\mapsto s$. Since each $\dot{z}^{\gamma\delta}$ is continuous on $[0,T]$, so is each $Z^{\gamma\delta}$ (claim 4 of the componentwise toolkit, the indefinite Riemann integral of a continuous integrand being continuous), and both are bounded there by the extreme value theorem, the ($\mathbf{1}$-modified) integrands built from these maps are bounded and product-measurable on a finite product measure.

We add the stopped objects. The family $\mathbf{1}\mathfrak{s}^\gamma=(\mathbf{1}\mathfrak{s}^\gamma_t)_{t\in[0,T]}$ is progressively measurable with respect to $(\mathcal{F}^{\mathrm{sys}}_t)_{t\in[0,T]}$ with every path right-continuous in the sequential sense of the progressive measurability toolkit: indeed $\mathbf{1}\mathfrak{s}^\gamma_t=\sqrt{N}(\mathbf{1}\Sigma^\gamma_t-\mathbf{1}S^\gamma_t)$; the family $\mathbf{1}\Sigma^\gamma$ is progressively measurable with every path right-continuous in the sequential sense by claim 2 of the stopped covariation lemma; the family $\mathbf{1}S^\gamma$ is adapted ($S^\gamma_t$ is a constant and $\mathbf{1}$ is $\mathcal{F}^{\mathrm{sys}}_0$-measurable, as noted in the preamble) with every path right-continuous — on $\Omega_0$ the path is $S^\gamma$ itself, continuous on $[0,T]$ by condition 1 of the mean-field trajectory pair, so path values converge along any sequence $s_j\to t$ in $[t,T]$, while off $\Omega_0$ the path vanishes identically — hence progressively measurable by claim 2 of the progressive measurability toolkit; and sums and scalar multiples of progressively measurable families are progressively measurable by claim 3 of the same toolkit, right-continuity of paths being preserved by the arithmetic of limits. Consequently, by claim 4 of the stopping-time toolkit (parts (i) and (iii), every path being right-continuous), the stopped family of $\mathbf{1}\mathfrak{s}^\gamma$ — which equals $(\mathbf{1}\hat{\mathfrak{s}}^\gamma_t)_{t\in[0,T]}$ pointwise, sampling being pointwise evaluation — is adapted with every path right-continuous, hence progressively measurable by claim 2 of the progressive measurability toolkit; in particular each $\mathbf{1}\hat{\mathfrak{s}}^\gamma_t$ is a random variable bounded by $2\sqrt{N}$, and $(s,\omega)\mapsto\mathbf{1}\hat{\mathfrak{s}}^\gamma_s(\omega)$ is product-measurable (claim 1 of that toolkit). Next, each $Z^{\gamma\delta}_{\min(t,\tau)}$ is a random variable bounded by the bound on $Z^{\gamma\delta}$: the function $\tau$ is measurable with respect to $\mathcal{F}^{\mathrm{sys}}_T$ and the Borel $\sigma$-algebra (claim 1 of the stopping-time toolkit), $x\mapsto\min(t,x)$ is sequentially continuous, and $u\mapsto Z^{\gamma\delta}_u$ is continuous on $[0,T]$, so the composition is measurable by two applications of the composition lemma. Therefore the functions named in part (a) — $\mathbf{1}\hat{\mathfrak{s}}^\gamma_t$, and the finite sums of products $\mathbf{1}\,\hat{\mathfrak{s}}_t\cdot Z_t\hat{\mathfrak{s}}_t$ and $\mathbf{1}\,\hat{\mathfrak{s}}_t\cdot Z_{\min(t,\tau)}\hat{\mathfrak{s}}_t$ — are bounded random variables, every expectation named in part (a) is finite and bounded in $s$, and measurability in $s$ of each follows from the Fubini theorem applied to the bounded product-measurable integrands $\mathbf{1}I_s\mathfrak{s}^\gamma_s\mathfrak{s}^\delta_s\,\dot{z}^{\gamma\delta}(s)$, $\mathbf{1}I_s\mathfrak{s}^\gamma_sg^\delta_s\,Z^{\gamma\delta}_s$, $I_s\,\mathbf{1}\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)$, and $\mathbf{1}\hat{\mathfrak{s}}^\gamma_s\hat{\mathfrak{s}}^\delta_s\,\dot{z}^{\gamma\delta}(s)$ on the finite product measure (the deterministic factors $\dot{z}^{\gamma\delta}(s)$ and $Z^{\gamma\delta}_s$, continuous in $s$, are product-measurable via composition with $(s,\omega)\mapsto s$). The expectation $\mathbb{E}[\mathfrak{s}_0\cdot Z_0\mathfrak{s}_0]$ named in part (a) is also finite: each $\mathfrak{s}^\gamma_0=\sqrt{N}(\Sigma^\gamma_0-S^\gamma_0)$ is a random variable bounded by $2\sqrt{N}$ ($\Sigma^\gamma_0$ is $\mathcal{F}^{\mathrm{sys}}_0$-measurable by part (iv) of the existence theorem, and $S^\gamma_0$ is a constant), so the pairing, a finite sum of products with the constants $Z^{\gamma\delta}_0$, is a bounded random variable. This proves (a).

Step 1 (stopped integral representation and cross moments). By part (b) of the martingale decomposition theorem and condition 2 of the mean-field trajectory pair (with claim 3 of the integral toolkit for the Riemann-Lebesgue agreement; the indicator $\mathbf{1}_{\Omega_0}$ in the decomposition's integral equals $1$ on $\Omega_0$), at every point of $\Omega_0$, for all $t$ and $\gamma$,

where each $M^\gamma$ is a square-integrable martingale with $M^\gamma_0=0$ at every point of $\Omega$ and $|M^\gamma_t|\le K_M$ everywhere (claim 2 of the stopped covariation lemma). Evaluating this representation at the time point $\min(t,\tau(\omega))\in[0,T]$ gives, at every $\omega\in\Omega_0$, for all $t$ and $\gamma$,

where $\hat{F}^\gamma_t$ is defined at every point of $\Omega$: at each $\omega$ the section $s\mapsto I_s(\omega)\,\mathbf{1}(\omega)\,g^\gamma_s(\omega)$ is bounded by $4\sqrt{N}(l-1)B$ (Step 0) and measurable on $[0,T]$ — the path of $I$ is measurable by claim 1 of the stopped-time-integral lemma; the path of $\mathbf{1}b^\gamma(\Sigma_\cdot,\alpha_\cdot)$ is measurable at every $\omega$ by part (a) of the decomposition theorem; the path $s\mapsto b^\gamma(S_s,A_s)$ is continuous by condition 2 of the mean-field trajectory pair, hence measurable by claim 3 of the Borel toolkit; and products and differences of measurable paths are measurable (the composition lemma). Moreover, at every $\omega\in\Omega_0$,

the first equality because $\mathbf{1}=1$ on $\Omega_0$ and the second by claim 2 of the stopped-time-integral lemma applied to the path of $g^\gamma$, measurable and bounded by $4\sqrt{N}(l-1)B$ on $\Omega_0$ as just noted. The sampled functions $\mathbf{1}\hat{\mathfrak{m}}^\gamma_t=\sqrt{N}\,\mathbf{1}M^\gamma_{\min(t,\tau)}$ are random variables bounded by $\sqrt{N}K_M$, by claim 4 of the stopped covariation lemma. We record four facts, for all $\gamma,\delta$ and $0\le s\le t\le T$.

(1a) If $X$ is a bounded random variable that is measurable with respect to the system filtration entry $\mathcal{F}^{\mathrm{sys}}_s$, then $\mathbb{E}[X\,\mathbf{1}(\hat{\mathfrak{m}}^\delta_t-\hat{\mathfrak{m}}^\delta_s)]=0$. Indeed, with $|X|\le c$, the dyadic truncations $X_n=2^{-n}\lfloor2^nX\rfloor$ (a finite sum $\sum_k k2^{-n}\mathbf{1}_{D_{n,k}}$ over the finitely many levels with $|k|2^{-n}\le c+1$, each $D_{n,k}\in\mathcal{F}^{\mathrm{sys}}_s$) satisfy $|X_n-X|\le2^{-n}$; the first identity of claim 4 of the stopped covariation lemma, applied with the pair of times $s\le t$ and the event $D_{n,k}\in\mathcal{F}^{\mathrm{sys}}_s$, gives $\mathbb{E}[\mathbf{1}_{D_{n,k}}\,\mathbf{1}(M^\delta_{\min(t,\tau)}-M^\delta_{\min(s,\tau)})]=0$ for each level set, hence $\mathbb{E}[X_n\,\mathbf{1}(\hat{\mathfrak{m}}^\delta_t-\hat{\mathfrak{m}}^\delta_s)]=0$ by linearity, and $|\mathbb{E}[(X-X_n)\,\mathbf{1}(\hat{\mathfrak{m}}^\delta_t-\hat{\mathfrak{m}}^\delta_s)]|\le2^{-n}\cdot2\sqrt{N}K_M\to0$.

(1b) $\mathbb{E}[\mathfrak{s}^\gamma_0\,\mathbf{1}\hat{\mathfrak{m}}^\delta_t]=0$: apply (1a) with $s=0$ and $X=\mathfrak{s}^\gamma_0$ (bounded by $2\sqrt{N}$; $\Sigma_0$ is $\mathcal{F}^{\mathrm{sys}}_0$-measurable by part (iv) of the existence theorem and $S_0$ is a constant), noting $\hat{\mathfrak{m}}^\delta_0=\sqrt{N}M^\delta_{\min(0,\tau)}=\sqrt{N}M^\delta_0=0$ at every point of $\Omega$.

(1c) $\mathbb{E}[I_s\,\mathbf{1}g^\gamma_s\,\mathbf{1}\hat{\mathfrak{m}}^\delta_t]=\mathbb{E}[I_s\,\mathbf{1}g^\gamma_s\,\mathbf{1}\hat{\mathfrak{m}}^\delta_s]$: apply (1a) with $X=I_s\,\mathbf{1}g^\gamma_s$, which is bounded by $4\sqrt{N}(l-1)B$ at every point of $\Omega$ (Step 0) and $\mathcal{F}^{\mathrm{sys}}_s$-measurable ($I_s$ is $\mathcal{F}^{\mathrm{sys}}_s$-measurable and $\mathbf{1}$ is $\mathcal{F}^{\mathrm{sys}}_0$-measurable, both as noted in the preamble; $\Sigma_s$ is $\mathcal{F}^{\mathrm{sys}}_s$-measurable by part (iv) of the existence theorem, $\alpha_s$ is $\mathcal{G}_s$-measurable by the same part, hence $\mathcal{F}^{\mathrm{sys}}_s$-measurable since $\mathcal{G}_s\subseteq\mathcal{F}^{\mathrm{sys}}_s$ by part (vii)(e), and $b^\gamma$ composed with them is measurable by the composition lemma; $b^\gamma(S_s,A_s)$ is a constant).

(1d) $\mathbb{E}[\mathbf{1}\hat{\mathfrak{m}}^\gamma_t\hat{\mathfrak{m}}^\delta_t]=\int_{[0,t]}\mathbb{E}[I_s\,\mathbf{1}\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)]\,ds$: this is the second identity of claim 4 of the stopped covariation lemma, with the pair of times $0\le t$ and $D=\Omega$ — the terms at time $0$ vanishing since $M^\gamma_{\min(0,\tau)}=M^\gamma_0=0$ everywhere — multiplied by $N$, together with the Fubini theorem to exchange $\mathbb{E}$ and the time integral of the product-measurable bounded integrand $I_s\,\mathbf{1}\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)$ (Step 0 and the preamble).

Step 2 (evolution of the stopped second-moment matrix). Fix $\gamma,\delta$ and set $\hat{\Psi}^{\gamma\delta}(t)=\mathbb{E}[\mathbf{1}\hat{\mathfrak{s}}^\gamma_t\hat{\mathfrak{s}}^\delta_t]$. Expanding the product of the two stopped three-term representations of Step 1 — valid at every point of $\Omega_0$, which suffices under the factor $\mathbf{1}$ — and taking expectations termwise (all nine terms are bounded random variables by Step 0 and Step 1, hence integrable, absorbing repeated factors via $\mathbf{1}^2=\mathbf{1}$):

the terms $\mathbb{E}[\mathbf{1}\mathfrak{s}^\gamma_0\hat{\mathfrak{m}}^\delta_t]$ and $\mathbb{E}[\mathbf{1}\hat{\mathfrak{m}}^\gamma_t\mathfrak{s}^\delta_0]$ vanishing by (1b). Now: $\mathbf{1}\hat{F}^\delta_t=\hat{F}^\delta_t$ at every point of $\Omega$ ($\hat{F}^\delta_t$ vanishes off $\Omega_0$, its integrand carrying the factor $\mathbf{1}$), so $\mathbb{E}[\mathbf{1}\mathfrak{s}^\gamma_0\hat{F}^\delta_t]=\int_{[0,t]}\mathbb{E}[I_s\,\mathbf{1}\mathfrak{s}^\gamma_0\,\mathbf{1}g^\delta_s]\,ds$ by the Fubini theorem (bounded product-measurable integrand: Step 0, the preamble, and part (iv) of the existence theorem for $\mathfrak{s}^\gamma_0$; the repeated factor $\mathbf{1}$ is absorbed by $\mathbf{1}^2=\mathbf{1}$), and symmetrically $\mathbb{E}[\mathbf{1}\hat{F}^\gamma_t\mathfrak{s}^\delta_0]=\int_{[0,t]}\mathbb{E}[I_s\,\mathbf{1}g^\gamma_s\,\mathbf{1}\mathfrak{s}^\delta_0]\,ds$. Pathwise at every $\omega\in\Omega$, the integration by parts lemma on $[0,t]$ with $u_0=v_0=0$ and densities $I_s\mathbf{1}g^\gamma_s$ and $I_s\mathbf{1}g^\delta_s$ — measurable and bounded on $[0,t]$ by Step 1, hence integrable there, the case $t=0$ being trivial — gives

(collecting the common factor $I_s\mathbf{1}$ from the two density terms), so $\mathbb{E}[\mathbf{1}\hat{F}^\gamma_t\hat{F}^\delta_t]=\int_{[0,t]}\mathbb{E}[I_s\,\mathbf{1}(g^\gamma_s\hat{F}^\delta_s+\hat{F}^\gamma_sg^\delta_s)]\,ds$ (Fubini; $|\hat{F}^\gamma_s|\le4\sqrt{N}(l-1)BT$ everywhere, and $(s,\omega)\mapsto\hat{F}^\gamma_s$ is product-measurable, being the indefinite time integral of the bounded product-measurable family $I\,\mathbf{1}g^\gamma$, which is progressively measurable with respect to the constant filtration $(\mathcal{F})_{t\in[0,T]}$ — the subsets of $[0,T]\times\Omega$ whose intersection with $[0,t]\times\Omega$ lies in $\mathcal{B}_{[0,t]}\otimes\mathcal{F}$ form a $\sigma$-algebra containing every measurable rectangle, hence all of $\mathcal{B}_{[0,T]}\otimes\mathcal{F}$ — so that claims 4 and 1 of the progressive measurability toolkit apply). Also $\mathbf{1}\hat{F}^\gamma_t\hat{\mathfrak{m}}^\delta_t=\int_{[0,t]}I_s\,\mathbf{1}g^\gamma_s\,\mathbf{1}\hat{\mathfrak{m}}^\delta_t\,ds$ pathwise at every $\omega$ (the constant-in-$s$ factor $\mathbf{1}\hat{\mathfrak{m}}^\delta_t$ moving inside the integral by linearity, with $\mathbf{1}^2=\mathbf{1}$), so by the Fubini theorem (the integrand $I_s\,\mathbf{1}g^\gamma_s\,\mathbf{1}\hat{\mathfrak{m}}^\delta_t$ is product-measurable and bounded by $4\sqrt{N}(l-1)B\cdot\sqrt{N}K_M$) and (1c),

and symmetrically for $\mathbb{E}[\mathbf{1}\hat{\mathfrak{m}}^\gamma_t\hat{F}^\delta_t]$. Combining with (1d) and collecting the integrands via $I_s\,(\mathfrak{s}^\delta_0+\hat{F}^\delta_s+\hat{\mathfrak{m}}^\delta_s)=I_s\,\hat{\mathfrak{s}}^\delta_s=I_s\,\mathfrak{s}^\delta_s$ on $\Omega_0$ — the first equality by Step 1, the second because $I_s(\omega)=1$ forces $s<\tau(\omega)$, whence $\min(s,\tau(\omega))=s$ and $\hat{\mathfrak{s}}^\delta_s(\omega)=\mathfrak{s}^\delta_s(\omega)$ —

with $\hat{\Psi}^{\gamma\delta}(0)=\mathbb{E}[\mathfrak{s}^\gamma_0\mathfrak{s}^\delta_0]$ (as $\min(0,\tau)=0$ pointwise and $P(\Omega_0)=1$) and each $\hat{\psi}^{\gamma\delta}$ bounded, and measurable by the same Fubini argument as in Step 0 applied to the bounded product-measurable integrands $\mathbf{1}I\,g^\gamma\mathfrak{s}^\delta$ and $I\,\mathbf{1}\Theta^{\gamma\delta}$.

Step 3 (weighting by $Z$: part (b)). Fix $t\in[0,T]$; the case $t=0$ is trivial (both sides of (b) equal $\mathbb{E}[\mathfrak{s}_0\cdot Z_0\mathfrak{s}_0]$, as $\min(0,\tau)=0$), so let $t>0$. Apply the integration by parts lemma on $[0,t]$ to $u=Z^{\gamma\delta}$ (with density $\dot{z}^{\gamma\delta}$, continuous, Riemann and Lebesgue integrals agreeing) and $v=\hat{\Psi}^{\gamma\delta}$ (with density $\hat{\psi}^{\gamma\delta}$, from Step 2, bounded and measurable there, hence integrable):

Sum over $\gamma,\delta\in\{1,\dots,l\}$. On the left, $\sum_{\gamma,\delta}Z^{\gamma\delta}_t\hat{\Psi}^{\gamma\delta}(t)=\mathbb{E}[\mathbf{1}\,\hat{\mathfrak{s}}_t\cdot Z_t\hat{\mathfrak{s}}_t]$ by linearity of the expectation, while $\sum_{\gamma,\delta}Z^{\gamma\delta}_0\hat{\Psi}^{\gamma\delta}(0)=\mathbb{E}[\mathfrak{s}_0\cdot Z_0\mathfrak{s}_0]$ by the value of $\hat{\Psi}^{\gamma\delta}(0)$ recorded in Step 2. In the integrand, $\sum_{\gamma,\delta}\dot{z}^{\gamma\delta}(s)\hat{\Psi}^{\gamma\delta}(s)=\mathbb{E}[\mathbf{1}\,\hat{\mathfrak{s}}_s\cdot\dot{z}(s)\hat{\mathfrak{s}}_s]$, while by the symmetry of $Z_s$ and relabeling of the summation indices,

and the remaining term is $\sum_{\gamma,\delta}Z^{\gamma\delta}_s\,\mathbb{E}[I_s\,\mathbf{1}\Theta^{\gamma\delta}(\Sigma_s,\alpha_s)]$. Since $I_s=\mathbf{1}_{\{s<\tau\}}$, this is exactly the displayed identity of part (b).

Step 4 (part (c)). Fix $t\in[0,T]$; for $t=0$ both sides of (c) equal $\mathbb{E}[\mathfrak{s}_0\cdot Z_0\mathfrak{s}_0]$, so let $t>0$. Since $Z^{\gamma\delta}_u=Z^{\gamma\delta}_0+\int_0^u\dot{z}^{\gamma\delta}(s)\,ds$ for every $u\in[0,T]$, with the Riemann and Lebesgue integrals agreeing (claim 3 of the integral toolkit), claim 2 of the stopped-time-integral lemma — applied at each $\omega$ to the deterministic family $(s,\omega)\mapsto\dot{z}^{\gamma\delta}(s)$, whose paths are continuous, hence measurable by claim 3 of the Borel toolkit, and bounded by the extreme value theorem — gives at every $\omega$

the last equality by linearity. Multiply by the random variable $\mathbf{1}\hat{\mathfrak{s}}^\gamma_t\hat{\mathfrak{s}}^\delta_t$, constant in $s$, moving it inside the integral by linearity, and note: at every $\omega$ and every $s\in[0,t]$ with $1-I_s(\omega)=1$ — that is, $\tau(\omega)\le s$ — one has $\min(t,\tau(\omega))=\tau(\omega)=\min(s,\tau(\omega))$, so $\hat{\mathfrak{s}}^\gamma_t(\omega)=\hat{\mathfrak{s}}^\gamma_s(\omega)$ and $\hat{\mathfrak{s}}^\delta_t(\omega)=\hat{\mathfrak{s}}^\delta_s(\omega)$. The integrands $(1-I_s)\,\mathbf{1}\hat{\mathfrak{s}}^\gamma_t\hat{\mathfrak{s}}^\delta_t\,\dot{z}^{\gamma\delta}(s)$ and $(1-I_s)\,\mathbf{1}\hat{\mathfrak{s}}^\gamma_s\hat{\mathfrak{s}}^\delta_s\,\dot{z}^{\gamma\delta}(s)$ therefore coincide pointwise, giving at every $\omega$

Take expectations, exchanging $\mathbb{E}$ with the time integral by the Fubini theorem (the integrand is product-measurable by Step 0 and claim 1 of the stopped-time-integral lemma, and bounded by $4N$ times the bound on $\dot{z}^{\gamma\delta}$), and sum over $\gamma,\delta$:

where we split $\mathbb{E}[\mathbf{1}(1-I_s)\hat{\mathfrak{s}}^\gamma_s\hat{\mathfrak{s}}^\delta_s\dot{z}^{\gamma\delta}(s)]=\mathbb{E}[\mathbf{1}\hat{\mathfrak{s}}^\gamma_s\hat{\mathfrak{s}}^\delta_s\dot{z}^{\gamma\delta}(s)]-\mathbb{E}[\mathbf{1}I_s\hat{\mathfrak{s}}^\gamma_s\hat{\mathfrak{s}}^\delta_s\dot{z}^{\gamma\delta}(s)]$ by linearity and used $I_s\hat{\mathfrak{s}}^\gamma_s=I_s\mathfrak{s}^\gamma_s$ (Step 2). Subtracting this display from the identity of part (b) — the difference of the two time integrals being the time integral of the difference, by linearity, all integrands being bounded measurable functions of $s$ by part (a) — and combining the two remaining expectations in the integrand into one by linearity yields exactly the displayed identity of part (c). $\blacksquare$