Proof of Almost-Everywhere Mean-Square Limits of Cauchy Sequences of Admissible Controls

Throughout, $\lVert\cdot\rVert_{2}$ is the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product, and $\mathcal{B}_{[0,T]}$, $\lambda_{[0,T]}$ and the claims of the interval toolkit are used with $[a,b]=[0,T]$; we abbreviate $\lambda:=\lambda_{[0,T]}$. For admissible controls $\beta,\gamma$ write $g_{\beta,\gamma}(t):=\sum_{\kappa=1}^{k}\lVert\beta^{\kappa}_t-\gamma^{\kappa}_t\rVert_{2}^{2}$, so that $d(\beta,\gamma)^{2}$ is the Riemann integral of the continuous function $g_{\beta,\gamma}$, as recorded in the statement; by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, $g_{\beta,\gamma}$ is $\mathcal{B}_{[0,T]}$-measurable and $d(\beta,\gamma)^{2}=\int g_{\beta,\gamma}\,d\lambda$. Recall also from the triangle inequality of claim 2 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm that for square-integrable $U,V$ one has $\bigl|\lVert U\rVert_{2}-\lVert V\rVert_{2}\bigr|\le\lVert U-V\rVert_{2}$, applied to $U=(U-V)+V$ and $V=(V-U)+U$.

Step 0 (triangle inequality for $d$). Let $\beta,\gamma,\rho$ be admissible. Set $A(t):=g_{\beta,\rho}(t)^{1/2}$ and $B(t):=g_{\rho,\gamma}(t)^{1/2}$, with the nonnegative square root. These are continuous: $A^{2}$ is continuous and nonnegative, and for real $x\ge y\ge0$ one has $x^{1/2}-y^{1/2}\le(x-y)^{1/2}$, since squaring reduces this to $x+y-2(xy)^{1/2}\le x-y$, that is, to $y\le(xy)^{1/2}$, which holds because $y^{1/2}\le x^{1/2}$; hence $|A(t)-A(s)|\le|A(t)^{2}-A(s)^{2}|^{1/2}\to0$ as $s\to t$. For fixed $t$, consider the vectors $u,v\in\mathbb{R}^{k}$ with entries $u_{\kappa}=\lVert\beta^{\kappa}_t-\rho^{\kappa}_t\rVert_{2}$, $v_{\kappa}=\lVert\rho^{\kappa}_t-\gamma^{\kappa}_t\rVert_{2}$. The mean-square triangle inequality gives $\lVert\beta^{\kappa}_t-\gamma^{\kappa}_t\rVert_{2}\le u_{\kappa}+v_{\kappa}$, so, squaring and summing, $g_{\beta,\gamma}(t)\le\sum_{\kappa}(u_{\kappa}+v_{\kappa})^{2}$. The map $w\mapsto(\sum_{\kappa}w_{\kappa}^{2})^{1/2}$ is the Euclidean distance to the origin, so by Euclidean Distance is a Metric on $\mathbb{R}^n$ $(\sum_{\kappa}(u_{\kappa}+v_{\kappa})^{2})^{1/2}\le A(t)+B(t)$. Hence $g_{\beta,\gamma}\le(A+B)^{2}$ pointwise, and by monotonicity and linearity of the Lebesgue integral of nonnegative measurable functions (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) together with claims 3 and 4 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval,

that is, $d(\beta,\gamma)\le d(\beta,\rho)+d(\rho,\gamma)$.

Claim 1. Using the Cauchy property, choose natural numbers $n_1<n_2<\dots$ such that $d(\alpha^{(n)},\alpha^{(m)})\le2^{-j}$ for all $n,m\ge n_j$; in particular $d_j:=d(\alpha^{(n_j)},\alpha^{(n_{j+1})})\le2^{-j}$. Define

Each $G_j$ is continuous: by the first paragraph, $\bigl|\lVert\alpha^{(n_j),\kappa}_t-\alpha^{(n_{j+1}),\kappa}_t\rVert_{2}-\lVert\alpha^{(n_j),\kappa}_s-\alpha^{(n_{j+1}),\kappa}_s\rVert_{2}\bigr|\le\lVert\alpha^{(n_j),\kappa}_t-\alpha^{(n_j),\kappa}_s\rVert_{2}+\lVert\alpha^{(n_{j+1}),\kappa}_t-\alpha^{(n_{j+1}),\kappa}_s\rVert_{2}$, which tends to $0$ as $s\to t$ by the mean-square continuity required by Admissible Control for the Linear-Gaussian State-Observation Model. Expanding the square of a sum and using $2xy\le x^{2}+y^{2}$ gives $G_j^{2}\le k\,g_{\alpha^{(n_j)},\alpha^{(n_{j+1})}}$ pointwise, so by claims 3 and 4 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval,

Let $H_J:=\sum_{j\le J}G_j$ and $H:=\sup_JH_J$, a function with values in $[0,\infty]$, equal at each $t$ to $\liminf_JH_J(t)$ and hence measurable by Fatou's Lemma. By the monotone convergence theorem and linearity (Linearity and Monotonicity of the Lebesgue Integral),

Let $N:=\{t:H(t)=\infty\}$. For every real $M>0$, the simple function $M\mathbf{1}_{\{H\ge M\}}$ is at most $H$, so $M\,\lambda(\{H\ge M\})\le\int H\,d\lambda$ by Simple Function and Its Integral and Lebesgue Integral of a Nonnegative Measurable Function. Since $N\subseteq\{H\ge M\}$, additivity of the measure gives $\lambda(N)\le\lambda(\{H\ge M\})$ (the difference set having nonnegative measure), so $\lambda(N)\le M^{-1}\int H\,d\lambda$ for every $M>0$, whence $\lambda(N)=0$. Set $D:=[0,T]\setminus N\in\mathcal{B}_{[0,T]}$, a co-null set.

Fix $t\in D$ and $\kappa$. The partial sums of $\sum_j\lVert\alpha^{(n_j),\kappa}_t-\alpha^{(n_{j+1}),\kappa}_t\rVert_{2}$ are nondecreasing and bounded by $H(t)<\infty$, hence have a least upper bound $S$ by the least upper bound property, and converge to it. Given $\varepsilon>0$ pick $i_0$ with $S_{i_0}>S-\varepsilon$, where $S_i$ denotes the $i$-th partial sum; then for $j>i>i_0$, the mean-square triangle inequality gives $\lVert\alpha^{(n_i),\kappa}_t-\alpha^{(n_j),\kappa}_t\rVert_{2}\le\sum_{r=i}^{j-1}\lVert\alpha^{(n_r),\kappa}_t-\alpha^{(n_{r+1}),\kappa}_t\rVert_{2}=S_{j-1}-S_{i-1}\le S-S_{i_0}<\varepsilon$. Thus $(\alpha^{(n_j),\kappa}_t)_j$ is Cauchy in mean square. By condition (ii) of Admissible Control for the Linear-Gaussian State-Observation Model, for each $j$ choose a $\mathcal{G}_t$-measurable square-integrable random variable almost surely equal to $\alpha^{(n_j),\kappa}_t$; almost sure equality leaves all mean-square distances unchanged, so these versions form a mean-square Cauchy sequence of $\mathcal{G}_t$-measurable random variables. By Mean-Square Completeness of Square-Integrable Random Variables (Riesz-Fischer) applied with the sub-$\sigma$-algebra $\mathcal{G}_t$, there is a $\mathcal{G}_t$-measurable square-integrable $\beta^{\kappa}_t$ with $\lVert\alpha^{(n_j),\kappa}_t-\beta^{\kappa}_t\rVert_{2}\to0$. For $t\notin D$ set $\beta^{\kappa}_t:=0$, which is $\mathcal{G}_t$-measurable. This proves claim 1.

Claim 2. Let $\beta$, $D$, and $(n_j)$ be as hypothesized, and fix $n$. For $t\in D$ and each $\kappa$, the triangle inequality gives $\bigl|\lVert\alpha^{(n),\kappa}_t-\beta^{\kappa}_t\rVert_{2}-\lVert\alpha^{(n),\kappa}_t-\alpha^{(n_j),\kappa}_t\rVert_{2}\bigr|\le\lVert\alpha^{(n_j),\kappa}_t-\beta^{\kappa}_t\rVert_{2}\to0$, so $\lVert\alpha^{(n),\kappa}_t-\alpha^{(n_j),\kappa}_t\rVert_{2}\to\lVert\alpha^{(n),\kappa}_t-\beta^{\kappa}_t\rVert_{2}$ as $j\to\infty$; squares of convergent real sequences converge to the square of the limit, since $|x_j^{2}-x^{2}|=|x_j-x|\,|x_j+x|$ with the second factor bounded. Hence the function $f:=\mathbf{1}_D\sum_{\kappa}\lVert\alpha^{(n),\kappa}-\beta^{\kappa}\rVert_{2}^{2}$ satisfies: $f=0$ off $D$, and $f(t)=\lim_jg_{\alpha^{(n)},\alpha^{(n_j)}}(t)$ for $t\in D$, each $g_{\alpha^{(n)},\alpha^{(n_j)}}$ being measurable and nonnegative. By claim 6 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, $f$ is $\mathcal{B}_{[0,T]}$-measurable. By Fatou's Lemma applied to the sequence $\bigl(g_{\alpha^{(n)},\alpha^{(n_j)}}\mathbf{1}_D\bigr)_j$, whose pointwise liminf is $f$ (off $D$ all terms vanish),

using claim 6 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval to drop the indicator and claim 3 there to identify the integral. By Step 0 the right side is finite (bound $d(\alpha^{(n)},\alpha^{(n_j)})\le d(\alpha^{(n)},\alpha^{(n_1)})+d(\alpha^{(n_1)},\alpha^{(n_j)})$, the last terms being bounded by the Cauchy property). Finally, given $\varepsilon>0$ choose $N$ with $d(\alpha^{(n)},\alpha^{(m)})<\varepsilon$ for $n,m\ge N$; since $n_j\ge N$ for large $j$, for every $n\ge N$ the right side is at most $\varepsilon^{2}$. This proves claim 2.

Claim 3. Let $(\beta,D)$ and $(\beta',D')$ be as hypothesized, with subsequences $(n_j)$ and $(n'_j)$. By claim 2, $u_n:=\mathbf{1}_D\sum_{\kappa}\lVert\alpha^{(n),\kappa}-\beta^{\kappa}\rVert_{2}^{2}$ and $u'_n:=\mathbf{1}_{D'}\sum_{\kappa}\lVert\alpha^{(n),\kappa}-\beta'^{\kappa}\rVert_{2}^{2}$ are measurable with $\int u_n\,d\lambda\to0$ and $\int u'_n\,d\lambda\to0$. Define $w:=\mathbf{1}_{D\cap D'}\sum_{\kappa}\lVert\beta^{\kappa}-\beta'^{\kappa}\rVert_{2}^{2}$. For $t\in D\cap D'$, as in claim 2, $\lVert\beta^{\kappa}_t-\beta'^{\kappa}_t\rVert_{2}=\lim_j\lVert\alpha^{(n_j),\kappa}_t-\beta'^{\kappa}_t\rVert_{2}$, so $w$ vanishes off $D\cap D'$ and is the pointwise limit on $D\cap D'$ of the measurable functions $\mathbf{1}_{D\cap D'}\sum_{\kappa}\lVert\alpha^{(n_j),\kappa}-\beta'^{\kappa}\rVert_{2}^{2}$ (measurable by the indicator-preimage argument of claim 6 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, multiplying $u'_{n_j}$ by $\mathbf{1}_D$); hence $w$ is measurable by claim 6 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. The mean-square triangle inequality and $(x+y)^{2}\le2x^{2}+2y^{2}$ give $w\le2u_n+2u'_n$ pointwise for every $n$, so by monotonicity and linearity (Linearity and Monotonicity of the Lebesgue Integral), $\int w\,d\lambda\le2\int u_n\,d\lambda+2\int u'_n\,d\lambda\to0$, whence $\int w\,d\lambda=0$. By claim 5 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, $\lambda(\{w>0\})=0$. Set $D'':=(D\cap D')\setminus\{w>0\}$; its complement in $[0,T]$ is a union of three $\lambda$-null sets, hence null by the subadditivity argument in claim 5 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. For $t\in D''$ and each $\kappa$: $\lVert\beta^{\kappa}_t-\beta'^{\kappa}_t\rVert_{2}=0$, and then $P(|\beta^{\kappa}_t-\beta'^{\kappa}_t|\ge1/n)\le n^{2}\,\mathbb{E}[(\beta^{\kappa}_t-\beta'^{\kappa}_t)^{2}]=0$ for every $n$ by Markov's inequality (Markov's and Chebyshev's Inequalities) applied to the square, so $\beta^{\kappa}_t=\beta'^{\kappa}_t$ almost surely. $\square$