= Solution
The <previsible sigma-algebra> $\mathcal P$ on $\Omega\times[0,\infty)$ is generated by left-continuous adapted processes. Equivalently it is generated by $A\times\{0\}$ with $A\in\mathcal F_0$ and $A\times(s,t]$ with $A\in\mathcal F_s$, $0\le s<t$. A <previsible process> is a process measurable for this sigma-algebra; <predictable> is a synonym.
Use the zero-starting convention for $\mathcal M_c^2$: continuous <martingales> with $M_0=0$ and $\sup_t\mathbb E M_t^2<\infty$. For such an <L2-bounded continuous martingale>, the <square-integrable stochastic integrand> space is
$$
\boxed{L^2(M)=\left\{H\text{ previsible}:\ \|H\|_{L^2(M)}^2=\mathbb E\int_0^\infty H_s^2\,d[M]_s<\infty\right\}.}
$$
Identify processes that agree almost everywhere for the <quadratic-variation measure> $\nu_M(d\omega,ds)=\mathbb P(d\omega)\,d[M]_s$. This makes the displayed norm an actual Hilbert-space norm on equivalence classes. On a fixed horizon $[0,T]$, replace infinity by $T$; this is the same construction for continuous <martingales> square-integrable on that horizon. Nonzero initial values may be subtracted because the integral depends only on increments.
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