 Solution

Let ${\mathcal{M}}_c^2$ be the continuous $L^2$-bounded martingales starting at zero, modulo indistinguishability, with norm $\Vert N{\Vert }_{{\mathcal{M}}^2}=\Vert N_{\infty }{\Vert }_2$. For a fixed $M\in {\mathcal{M}}_c^2$, define a finite measure on $\mathcal{P}$ byand let $L^2(M)=L^2(\mathcal{P},{\nu }_M)$. The Itô isometry is the isometric extensionsatisfying

For the simple process in part b, orthogonality gives the sum there. Conditional on ${\mathcal{F}}_{t_i}$, the martingale identity for $M^2-[M]$ givesSumming proves the isometry. Part c then supplies the unique extension to all of $L^2(M)$.