Adapted Mean-Square Continuous Processes are Ito Integrable

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P)$ be a filtered probability space, let $(M,\rho)$ be an It^{o} integrator of intensity type with respect to $(\mathcal{F}_t)_{t\ge0}$, let $\mathbb{R}$ be the real numbers, let $T>0$ be real, and let $\lambda$ denote Lebesgue measure. A real-valued function defined on a subinterval $I$ of $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Let $(H_t)_{t\in[0,T]}$ be a family of square-integrable random variables such that each $H_t$ is $\mathcal{F}_t$-measurable and $(H_t)_{t\in[0,T]}$ is mean-square continuous on $[0,T]$.

1. For each natural number $k\ge1$, define $H^{k}_{t}=H_{iT/k}$ for $t\in(iT/k,(i+1)T/k]$, $0\le i\le k-1$. Then each $H^{k}$ is a simple adapted process on $(0,T]$, and $(H^{k})_{k}$ is an approximating sequence for $(H_t)_{t\in(0,T]}$ in the sense of Existence and Uniqueness of the Mean-Square Extension of the Elementary Stochastic Integral. In particular, $(H_t)_{t\in(0,T]}$ is It^{o} integrable on $(0,T]$.

2. The function $t\mapsto\mathbb{E}[H_t^{2}]$ is continuous on $[0,T]$, hence measurable when extended by $0$, and for every $t\in(0,T]$,

3. In particular, let $f:[0,T]\to\mathbb{R}$ be continuous. Then the family of constant random variables $(f(t))_{t\in[0,T]}$ satisfies the hypotheses above for every filtration, so for every $t\in(0,T]$ the family $(f(u))_{u\in(0,t]}$ is It^{o} integrable on $(0,t]$, with