Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral

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 $T>0$ be real, let $H$ and $G$ be simple adapted processes on $(0,T]$, and let $a,b$ be real numbers. Write $\int_0^T H_t\,dM_t$ for the elementary stochastic integral and $\lambda$ for Lebesgue measure.

1. (Linearity) The family $aH+bG=(aH_t+bG_t)_{t\in(0,T]}$ is a simple adapted process on $(0,T]$, and

2. (Square-integrability and mean zero) $\int_0^T H_t\,dM_t$ is square-integrable and its expectation is

3. (Isometry and polarization) The functions $t\mapsto\mathbb{E}[H_tG_t]$ and $t\mapsto\mathbb{E}[H_t^{2}]$ on $(0,T]$ are step functions (constant on the intervals of a common representation), hence measurable when extended by $0$ to $\mathbb{R}$, the products with $\rho$ are integrable on $(0,T]$, and

in particular, taking $G=H$,