= Solution
A <simple predictable process> has the form $H_s=\sum_{j=0}^{m-1}h_j\mathbf1_{(t_j,t_{j+1}]}(s)$, where $0=t_0<\cdots<t_m<\infty$ are deterministic and $h_j$ is bounded and $\mathcal F_{t_j}$-measurable. A possible $\mathcal F_0$-measurable value at zero is irrelevant to this <Itô integral>. Define
$$
\boxed{(H\mathbin\cdot M)_t=\sum_{j=0}^{m-1}h_j\bigl(M_{t\wedge t_{j+1}}-M_{t\wedge t_j}\bigr).}
$$
Each coefficient is known before the corresponding increment. The <conditional expectation> of each subsequent increment is zero, so the result is a continuous <martingale> starting at zero. It is an <L2-bounded continuous martingale> because it is a finite sum of bounded coefficients times stopped increments of an <L2-bounded continuous martingale>. This definition is independent of the chosen subdivision: splitting an interval simply splits its increment into a telescoping sum. General admissible <predictable processes> are then integrated by completion using the <Itô isometry>; unbounded step coefficients are allowed when the weighted $L^2$ condition in the <quadratic-variation measure> in the next part holds.
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