Consider a \reftext{def:linear-gaussian-state-observation-model-2026a}{linear-Gaussian state-observation model} on $[0,T]$ with observation $\sigma$-algebras $\mathcal{G}_t$, let $k\ge1$ be a \reftext{def:natural-numbers-2026a}{natural number}, and let $(\alpha^{(n)})_{n\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{sequence} of \reftext{def:admissible-control-2026a}{admissible controls with values in $\mathbb{R}^{k}$} for the model. For admissible controls $\beta,\gamma$ define, with the mean-square norm $\lVert\cdot\rVert_{2}$ of \ref{def:square-integrable-mean-square-2026a},

a \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integral} of a function of $t$ that is \reftext{def:continuity-closed-interval-c54-2026b}{continuous}, and hence integrable by \ref{lem:continuous-implies-riemann-integrable-c54-2026b}: the componentwise difference family is \reftext{def:mean-square-continuous-process-2026a}{mean-square continuous} by claim 1 of \ref{lem:mean-square-riemann-integral-properties-2026a}, so continuity of the integrand follows from claim 2 of \ref{lem:expected-quadratic-form-2026a} applied to that family and the \reftext{def:identity-matrix-2026a}{identity matrix}. Suppose the sequence is \textbf{Cauchy for $d$}: for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $d(\alpha^{(n)},\alpha^{(m)})<\varepsilon$ for all $n,m\ge N$. Adopt the notation $\mathcal{B}_{[0,T]}$, $\lambda_{[0,T]}$ of \reftext{lem:interval-lebesgue-toolkit-2026a}{the restricted Lebesgue measure on $[0,T]$}, and call $D\in\mathcal{B}_{[0,T]}$ \textbf{co-null} if $\lambda_{[0,T]}([0,T]\setminus D)=0$. Then:

\textbf{1. (Existence of an almost-everywhere limit)} There exist a co-null set $D\in\mathcal{B}_{[0,T]}$, natural numbers $n_1<n_2<n_3<\dots$, and a \reftext{def:family-subfamily-subsets-set-2026a}{family} $\beta=(\beta_t)_{t\in[0,T]}$ of tuples $\beta_t=(\beta^{1}_t,\dots,\beta^{k}_t)$ of \reftext{def:square-integrable-mean-square-2026a}{square-integrable} random variables such that: $\beta^{\kappa}_t=0$ for every $t\notin D$ and every $\kappa$; for every $t\in[0,T]$ and $\kappa$ the random variable $\beta^{\kappa}_t$ is $\mathcal{G}_t$-measurable (preimages of \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel sets} belong to $\mathcal{G}_t$); and for every $t\in D$ and every $\kappa$ the real sequence $\bigl(\lVert\alpha^{(n_j),\kappa}_t-\beta^{\kappa}_t\rVert_{2}\bigr)_{j}$ has \reftext{def:limit-sequence-real-c54-2026a}{limit} $0$.

\textbf{2. (Integrated convergence of the full sequence)} Let $\beta$ be any family of tuples of square-integrable random variables and $D\in\mathcal{B}_{[0,T]}$ a co-null set such that, for some natural numbers $n_1<n_2<\dots$, every $t\in D$, and every $\kappa$: $\lVert\alpha^{(n_j),\kappa}_t-\beta^{\kappa}_t\rVert_{2}\to0$ as $j\to\infty$. Then for every $n$ the function

is $\mathcal{B}_{[0,T]}$-measurable with finite \reftext{def:lebesgue-integral-nonnegative-2026a}{Lebesgue integral}, and these integrals tend to $0$ as $n\to\infty$ along the full sequence.

\textbf{3. (Uniqueness almost everywhere)} If $(\beta,D)$ and $(\beta',D')$ both satisfy the hypotheses of claim 2 (with possibly different subsequences), then there is a co-null set $D''\in\mathcal{B}_{[0,T]}$ such that for every $t\in D''$ and every $\kappa$: $\beta^{\kappa}_t=\beta'^{\kappa}_t$ \reftext{def:almost-surely-2026a}{almost surely}.