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

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $a<b$ be real numbers.

1. (Uniqueness) Let $(H_t)_{t\in[a,b]}$ be any family of square-integrable random variables on $(\Omega,\mathcal{F},P)$. If $I$ and $I'$ are both mean-square Riemann integrals of $(H_t)_{t\in[a,b]}$ over $[a,b]$, then $I=I'$ almost surely.

2. (Existence) If $(H_t)_{t\in[a,b]}$ is mean-square continuous on $[a,b]$, then it is mean-square Riemann integrable on $[a,b]$. In particular, for all $s,t\in[a,b]$ with $s<t$, the restricted family $(H_u)_{u\in[s,t]}$ is mean-square continuous on $[s,t]$ and hence mean-square Riemann integrable on $[s,t]$, so that all the integrals $\int_s^t H_u\,du$ of Mean-Square Riemann Integral of a Family of Random Variables exist.

3. (Measurability) In the situation of claim 2, let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$ such that every $H_t$ ($t\in[a,b]$) is $\mathcal{G}$-measurable, that is, $H_t^{-1}(B)\in\mathcal{G}$ for every Borel set $B$. Then some mean-square Riemann integral of $(H_t)_{t\in[a,b]}$ over $[a,b]$ is $\mathcal{G}$-measurable, and likewise over every subinterval $[s,t]\subseteq[a,b]$ with $s<t$; by claim 1, every mean-square Riemann integral over the same interval is then almost surely equal to the $\mathcal{G}$-measurable one.