= Solution
Let $0=t_0<t_1<\cdots<t_m<\infty$ and let $\xi_j$ be bounded $\mathcal F_{t_j}$-<measurable> <random variables>. For the bounded <simple predictable process>
$$
H_s=\sum_{j=0}^{m-1}\xi_j\mathbf1_{(t_j,t_{j+1}]}(s),
$$
the <Itô integral> against a <continuous martingale> $M\in\mathcal H^{2,c}$ is defined by the increment sum
$$
\boxed{(H\mathbin\cdot M)_t=\int_0^tH_s\,dM_s
:=\sum_{j=0}^{m-1}\xi_j\bigl(M_{t\wedge t_{j+1}}-M_{t\wedge t_j}\bigr).}
$$
The same definition applies to <elementary predictable processes with stopping-time intervals>, replacing $t_j$ by ordered <stopping times> $T_j$ and taking bounded $\mathcal F_{T_j}$-<measurable> coefficients. Refinement of the intervals leaves the increment sum unchanged. The initial value of the <integral> is zero, and its <martingale> property follows from the <conditional expectation> identity for subsequent increments of $M$.
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