Proposition

Let $M=(M_{t})_{t\ge 0}$ be a right continuous $L^{2}$-martingale. We have that the Itô Isometry holds $\forall X\in L^{2}(\mathbb{R}^{+}\times\Omega,\mathscr{P},\mu_{M})$ where $$E\left[ \left( \int\limits X \, dM \right)^{2} \right]=\int\limits _{\mathbb{R}^{+}\times\Omega}X^{2} \, d\mu_{M}$$or $$\left\lVert \int\limits X \, dM \right\rVert _{L^{2}(\Omega,\mathcal{F},P)}=\lVert X \rVert _{L^{2}(\mathbb{R}^{+}\times\Omega,\mathscr{P},\mu_{m})}$$