Let $(X,\mathcal{B}(X),(\mathcal{A}_n)_{n=1}^N,\mu)$ be a filtered probability space with $\mathcal{A}_N=\mathcal{B}(X)$ and let $H$ be the space of $\mathcal{A}_{\cdot}$-adapted martingales $m_{\cdot}$ satisfying: $$ \mathbb{E}\left[\sum_{n=1}^N (m_n)^2dt\right]<\infty. $$ The space $H$ is a separable Hilbert space with inner product $$ \langle m_{\cdot},m'_{\cdot}\rangle\triangleq \mathbb{E}\left[\sum_{n=1}^N|m_nm'_n|dt\right]. $$

Where can I find an example of an explicit orthonormal basis for this space? I expect that, given an orthogonal basis $\{h^k\}_{k=1}^{\infty}$ of $L^2(\mathcal{B}(X))=L^2(\mathcal{A}_N)$ one can construct an orthogonal basis of $H$ using Doob's construction $$ h^{k}_n:=\mathbb{E}_{\mathbb{P}}\left[h_k\mid \mathcal{A}_n\right]. $$ However, is this true and, if so, can someone provide a reference to a body of literature for which this is used?



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