Mean-Square Riemann Integral of a Family of Random Variables

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $a<b$ be real numbers, and let $(H_t)_{t\in[a,b]}$ be a family of square-integrable random variables on $(\Omega,\mathcal{F},P)$, with $\lVert\cdot\rVert_{2}$ the mean-square norm of that definition.

Let $P'=(x_0,x_1,\dots,x_n)$ be a partition of $[a,b]$, with mesh $|P'|$ as defined there, together with tags $\tau_i\in[x_{i-1},x_i]$ ($1\le i\le n$) forming a tagged partition. The associated mean-square Riemann sum of $(H_t)_{t\in[a,b]}$ is the random variable

which is square-integrable by the closure properties recorded in Square-Integrable Random Variables and the Mean-Square Inner Product.

A square-integrable random variable $I$ on $(\Omega,\mathcal{F},P)$ is a mean-square Riemann integral of $(H_t)_{t\in[a,b]}$ over $[a,b]$ if for every real $\varepsilon>0$ there is a real $\delta>0$ such that every mean-square Riemann sum $S$ associated with a tagged partition of $[a,b]$ of mesh less than $\delta$ satisfies

The family $(H_t)_{t\in[a,b]}$ is mean-square Riemann integrable on $[a,b]$ if such an $I$ exists.

Notation. When the family is mean-square Riemann integrable on $[a,b]$, the symbol

denotes a mean-square Riemann integral of $(H_t)_{t\in[a,b]}$ over $[a,b]$. Every statement containing this symbol presupposes that the family is mean-square Riemann integrable on $[a,b]$ and abbreviates the assertion that the statement holds for every random variable $I$ that is a mean-square Riemann integral of the family over $[a,b]$.

Subintervals and degenerate intervals. For $s,t\in[a,b]$ with $s<t$, the symbol $\int_s^t H_u\,du$ denotes a mean-square Riemann integral of the restricted family $(H_u)_{u\in[s,t]}$ over $[s,t]$; its use presupposes that the restricted family is mean-square Riemann integrable on $[s,t]$, a hypothesis separate from integrability on $[a,b]$. For every $s\in[a,b]$ we set $\int_s^s H_u\,du=0$, and no meaning is assigned to $\int_s^t H_u\,du$ when $s>t$. Likewise, for every real-valued function $\varphi$ and every point $s$ of its domain, the degenerate-interval Riemann integral $\int_s^s\varphi(u)\,du$ is set equal to $0$; this convention is unconditional and may be used wherever such a degenerate integral occurs.