Definition

Let $M$ be a right continuous $L^{2}$-martingale. We denote by $$\Lambda^{2}(\mathscr{P},M)$$the set of all RVs $X:(\mathbb{R}^{+}\times\Omega,\mathscr{P})\to(\mathbb{R},\mathcal{B}(\mathbb{R}))$ that are measurable (i.e. $X$ is a predictable process) s.t. $$\mathbb{1}_{[0,t]}X\in L^{2}(\mathbb{R}^{+}\times\Omega,\mathscr{P},\mu_{M}),\quad t\ge 0$$