Proof of Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families

Throughout, $\lVert\cdot\rVert_{2}$ is the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product, and we use the triangle inequality for it from Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm. Mean-square Riemann sums, tags, and mesh are as in that definition.

Claim 1. Let $\varepsilon>0$, and let $\delta,\delta'>0$ be as in Mean-Square Riemann Integral of a Family of Random Variables for $I$ and for $I'$ respectively, for this $\varepsilon$. Choose a natural number $n$ with $(b-a)/n<\min(\delta,\delta')$ and let $S$ be the mean-square Riemann sum for the partition of $[a,b]$ into $n$ intervals of equal length with left endpoints as tags; its mesh is $(b-a)/n$. Then

Since $\varepsilon>0$ was arbitrary, $\lVert I-I'\rVert_{2}=0$, and by the null-equivalence statement of Square-Integrable Random Variables and the Mean-Square Inner Product, $I=I'$ almost surely.

Claim 2. By Uniform Mean-Square Continuity on a Compact Interval, $(H_t)_{t\in[a,b]}$ is uniformly mean-square continuous: given $\varepsilon>0$ there is $\delta>0$ such that $\lVert H_s-H_t\rVert_{2}<\varepsilon$ for all $s,t\in[a,b]$ with $|s-t|<\delta$.

Refinement estimate. Let $S$ be the mean-square Riemann sum of a tagged partition $P=(x_0,\dots,x_n)$ of $[a,b]$ with tags $\tau_1,\dots,\tau_n$ and mesh less than $\delta$, and let $S'$ be the mean-square Riemann sum of a tagged partition $P'=(y_0,\dots,y_N)$ with tags $\tau'_1,\dots,\tau'_N$, whose division points include all division points of $P$. Each interval $[y_{k-1},y_k]$ of $P'$ is contained in exactly one interval of $P$ (intervals of $P'$ meeting two intervals of $P$ would contain a division point of $P$ in their interior, impossible since those are division points of $P'$ too); write $\tau(k)$ for the tag of that interval of $P$. Both $\tau(k)$ and $\tau'_k$ lie in one interval of $P$, of length less than $\delta$, so $|\tau(k)-\tau'_k|<\delta$. Since the lengths $y_k-y_{k-1}$ of the intervals of $P'$ within one interval of $P$ add up to the length of that interval,

by the triangle inequality. Consequently, if $S_1,S_2$ are mean-square Riemann sums of two tagged partitions each of mesh less than $\delta$, comparing both with a tagged partition on the common refinement (all division points of both, with arbitrary tags) gives

Construction of the integral. For each natural number $n\ge1$ let $S_n$ be the mean-square Riemann sum for the partition of $[a,b]$ into $n$ equal intervals with left endpoints as tags; its mesh is $(b-a)/n$. The sequence $(S_n)_n$ is Cauchy in mean square in the sense of Mean-Square Completeness of Square-Integrable Random Variables (Riesz-Fischer): given $\eta>0$, apply the preceding paragraph with $\varepsilon=\eta/(2(b-a)+1)$ to obtain $\delta>0$; then for all $n,m$ with $(b-a)/n<\delta$ and $(b-a)/m<\delta$,

By Mean-Square Completeness of Square-Integrable Random Variables (Riesz-Fischer) there is a square-integrable random variable $I$ with $\lVert S_n-I\rVert_{2}\to0$.

Now let $\varepsilon'>0$ be given. Apply the refinement estimate with $\varepsilon=\varepsilon'/(3(b-a))$, giving $\delta>0$. For every mean-square Riemann sum $S$ of a tagged partition of mesh less than $\delta$ and every $n$ with $(b-a)/n<\delta$,

and letting $n\to\infty$ gives $\lVert S-I\rVert_{2}\le\tfrac{2}{3}\varepsilon'<\varepsilon'$. Hence $I$ is a mean-square Riemann integral of $(H_t)_{t\in[a,b]}$ over $[a,b]$.

For the last statement of claim 2: the restriction of a mean-square continuous family to $[s,t]\subseteq[a,b]$ is mean-square continuous on $[s,t]$ directly from Mean-Square Continuous Family of Random Variables, so the argument just given applies on $[s,t]$.

Claim 3. Write again $S_n=\sum_{i=1}^{n}H_{\tau_i}(x_i-x_{i-1})$ for the uniform left-tagged sums of the previous paragraph, so each $S_n$ is a finite sum of scalar multiples of the $\mathcal{G}$-measurable random variables $H_{\tau_i}$. Such combinations are $\mathcal{G}$-measurable. Indeed, for $\mathcal{G}$-measurable $X$ and real $c>0$, $\{cX>\lambda\}=\{X>\lambda/c\}\in\mathcal{G}$ (similarly for $c<0$ with the reversed inequality, and $c=0$ gives a constant), and for $\mathcal{G}$-measurable $X,Y$,

a countable union of members of $\mathcal{G}$; by the generator criterion applied on the measurable space $(\Omega,\mathcal{G})$, the sets $\{\cdot>\lambda\}$ ($\lambda\in\mathbb{R}$) suffice for $\mathcal{G}$-measurability. Hence each $S_n$ is $\mathcal{G}$-measurable, and Mean-Square Completeness of Square-Integrable Random Variables (Riesz-Fischer), applied with the sub-$\sigma$-algebra $\mathcal{G}$, produces a $\mathcal{G}$-measurable square-integrable $I$ with $\lVert S_n-I\rVert_{2}\to0$; by the argument of claim 2, this $I$ is a mean-square Riemann integral of the family over $[a,b]$. The same argument applies on every subinterval $[s,t]\subseteq[a,b]$ with $s<t$, since the restricted family satisfies the same hypotheses there; and by claim 1 any mean-square Riemann integral over the same interval is almost surely equal to the $\mathcal{G}$-measurable one. $\blacksquare$