What is Simple process, Itô integral?

A simple process is $H_t = \sum_{k=0}^{n-1}h_k\mathbf 1_{(t_k, t_{k+1}]}(t)$ with $0 = t_0 < \dots < t_n = T$ and each $h_k$ bounded and $\mathcal F_{t_k}$-measurable. Its Itô integral is $\int_0^t H_s\,dW_s = \sum_k h_k(W_{t_{k+1}\wedge t} - W_{t_k\wedge t})$. For an adapted $H$ with $\E\int_0^T H_s^2\,ds < \infty$, the Itô integral is the $L^2$ limit of the integrals of simple processes $H^{(n)}$ with $\E\int_0^T(H^{(n)}_s - H_s)^2\,ds \to 0$.