Proof of Integrals Against the Observation Process are Determined by the Observations

Fix the component index $i$ and abbreviate the modulus of continuity: by Continuity on a Closed Interval Implies Uniform Continuity choose $\omega_n\to0$ bounding $|f_{ij}(v)-f_{ij}(v')|$ for all entries $j$ and all $v,v'\in[0,t]$ with $|v-v'|\le t/n$. Norms and Cauchy-Schwarz are from Square-Integrable Random Variables and the Mean-Square Inner Product and Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm.

Claim 1. By claim 5 of Basic Properties of the Mean-Square Riemann Integral and the interval-splitting of Wiener integrals (Ito Integrable Process and the Ito Integral, claim 1 of Properties of the Ito Integral: Linearity, Isometry, Martingale Property, and Mean-Square Continuity), almost surely,

Hence the $i$-th component of the Riemann-Stieltjes sum splits into a time part $\sum_{p}\sum_jf_{ij}(x_{p-1})\int_{x_{p-1}}^{x_p}(\tilde EX)^{j}_v\,dv$ and a Wiener part.

Time part. By claims 1, 2, 5 of Basic Properties of the Mean-Square Riemann Integral (with the $n$-fold split of $\int_0^t$ obtained by induction from claim 5), the difference from $\int_0^t(f\tilde EX)^{i}_v\,dv$ equals $\sum_{p}\sum_j\int_{x_{p-1}}^{x_p}\bigl(f_{ij}(x_{p-1})-f_{ij}(v)\bigr)(\tilde EX)^{j}_v\,dv$, whose norm is at most $\omega_n\sum_j\int_0^t\lVert(\tilde EX)^{j}_v\rVert_2\,dv\to0$ by the norm bound (claim 4 there).

Wiener part. Fix $j'$. With $V^{jj'}_s:=\int_0^s\tilde\varepsilon_{jj'}\,dW^{j'}$, the relevant difference is

All terms are Wiener integrals of continuous functions against $W^{j'}$, members of the centered jointly Gaussian family of claims 2-3 of Wiener Integrals Against a Vector Brownian Motion are Jointly Gaussian, so $\mathbb{E}[D_n^{2}]$ is a finite bilinear combination of the covariances given by claim 3 there: increments $V^{jj'}_{x_p}-V^{jj'}_{x_{p-1}}$ and $V^{j''j'}_{x_q}-V^{j''j'}_{x_{q-1}}$ over disjoint cells $p\ne q$ are uncorrelated (evaluate the four $\min$ terms), while over the same cell $\operatorname{Cov}\bigl(V^{jj'}_{x_p}-V^{jj'}_{x_{p-1}},V^{j''j'}_{x_p}-V^{j''j'}_{x_{p-1}}\bigr)=\int_{x_{p-1}}^{x_p}\tilde\varepsilon_{jj'}\tilde\varepsilon_{j''j'}\,dv$ and $\operatorname{Cov}\bigl(V^{jj'}_{x_p}-V^{jj'}_{x_{p-1}},\int_0^t(f\tilde\varepsilon)_{ij'}\,dW^{j'}\bigr)=\int_{x_{p-1}}^{x_p}\tilde\varepsilon_{jj'}(f\tilde\varepsilon)_{ij'}\,dv$ (Additivity of the Riemann Integral on Adjacent Intervals for the interval splittings). Writing $s_n(v)=\sum_jf_{ij}(x_{p(v)-1})\tilde\varepsilon_{jj'}(v)$, where $p(v)$ is the index with $v\in(x_{p(v)-1},x_{p(v)}]$ (so $s_n$ depends on $i$ and $j'$), collecting the three groups of terms and completing the square in each cell yields

since $|s_n(v)-(f\tilde\varepsilon)_{ij'}(v)|=|\sum_j(f_{ij}(x_{p(v)-1})-f_{ij}(v))\tilde\varepsilon_{jj'}(v)|\le\tilde l\,\omega_n\max|\tilde\varepsilon|$, with $\max|\tilde\varepsilon|$ a bound on all entries of $\tilde\varepsilon$ (Extreme Value Theorem on a Compact Interval); monotonicity of the Riemann integral is via Agreement of the Riemann and Lebesgue Integrals for Continuous Functions on a Closed Interval and Linearity and Monotonicity of the Lebesgue Integral. Summing the two parts over the finitely many indices (triangle inequality) proves claim 1.

Claim 2. Each Riemann-Stieltjes sum is a finite linear combination of the values $u^{j}_{x_p}$ with $x_p\le t$, all $\mathcal{G}_t$-measurable. By claim 1 the sums converge in mean square to the component of $\int_0^tf\,du$, so that component lies in the closed mean-square span of the values $u^{j}_r$ ($r\le t$) — the second assertion — and, by claim 2 of The Closed Mean-Square Span of a Family of Random Variables applied with $\mathcal{G}=\mathcal{G}_t$, it is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable. $\blacksquare$