Predictable representation in a Rademacher filtration

If ${\mathcal{F}}_n$ is generated by independent symmetric signs ${\xi }_1,\dots ,{\xi }_n$, every martingale has predictable coefficients $B_n$ such thatOne may take $B_n=\mathbb{E}[M_n{\xi }_n\mid {\mathcal{F}}_{n-1}]$.

For a bounded stopping time $T$ and a square-integrable representation $M_n=M_0+{\sum }_{k\le n}{B}_k{\xi }_k$,The cross terms vanish because predictable multiples of distinct independent signs are orthogonal martingale differences.