Basic Properties of the Mean-Square Riemann Integral

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathbb{R}$ be the real numbers, let $a<b$ be real numbers, and let $(H_t)_{t\in[a,b]}$ and $(G_t)_{t\in[a,b]}$ be mean-square continuous families of square-integrable random variables on $(\Omega,\mathcal{F},P)$, with $\lVert\cdot\rVert_{2}$ and $\langle\cdot,\cdot\rangle_{2}$ the mean-square norm and inner product of that definition. A real-valued function on $[a,b]$ is called continuous on $[a,b]$ when it is continuous relative to $[a,b]$, both $[a,b]$ and the codomain $\mathbb{R}$ carrying the metric of the real line. All mean-square Riemann integrals below exist by Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families, all identities between random variables are almost sure identities, all integrals of real-valued functions are Riemann integrals of continuous functions, which exist by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, and degenerate intervals follow the conventions of Mean-Square Riemann Integral of a Family of Random Variables.

1. (Linearity) For all real $\alpha,\beta$, the family $(\alpha H_t+\beta G_t)_{t\in[a,b]}$ is mean-square continuous on $[a,b]$ and

2. (Continuous deterministic factors) Let $c:[a,b]\to\mathbb{R}$ be continuous. Then the family $(c(t)H_t)_{t\in[a,b]}$ is mean-square continuous on $[a,b]$. Moreover, for every square-integrable random variable $Z$, the family $(c(t)Z)_{t\in[a,b]}$ is mean-square continuous on $[a,b]$ and

3. (Inner products and covariances) For every square-integrable random variable $Z$, the function $t\mapsto\langle Z,H_t\rangle_{2}=\mathbb{E}[ZH_t]$ is continuous on $[a,b]$ and

Likewise the function $t\mapsto\operatorname{Cov}(Z,H_t)$ is continuous, with the covariance of square-integrable random variables, and

Taking $Z=1$ in the first identity, with the expectation: $\mathbb{E}\bigl[\int_a^b H_t\,dt\bigr]=\int_a^b\mathbb{E}[H_t]\,dt$, the function $t\mapsto\mathbb{E}[H_t]$ being continuous.

4. (Norm bound) The function $t\mapsto\lVert H_t\rVert_{2}$ is continuous on $[a,b]$ and

5. (Additivity) For every $r\in[a,b]$,

6. (Mean-square continuity of the indefinite integral) For all $s,t\in[a,b]$ with $s<t$,

the maximum existing by Extreme Value Theorem on a Closed Real Interval; for $s=t$ all three quantities are $0$ by the degenerate-interval conventions of Mean-Square Riemann Integral of a Family of Random Variables. Consequently, for every fixed choice of versions, the family $\bigl(\int_a^t H_u\,du\bigr)_{t\in[a,b]}$ is mean-square continuous on $[a,b]$.