Simple Adapted Process

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P)$ be a filtered probability space and let $T>0$ be a real number.

A simple adapted process on $(0,T]$ is a family $H=(H_t)_{t\in(0,T]}$ of random variables on $(\Omega,\mathcal{F},P)$ for which there exist a natural number $n\ge1$, real numbers $0=t_0<t_1<\dots<t_n=T$, and square-integrable random variables $\xi_0,\dots,\xi_{n-1}$, with each $\xi_i$ being $\mathcal{F}_{t_i}$-measurable (that is, $\xi_i^{-1}(B)\in\mathcal{F}_{t_i}$ for every Borel set $B$), such that

The data $\bigl((t_i)_{i=0}^{n},(\xi_i)_{i=0}^{n-1}\bigr)$ is called a representation of $H$.