Recovering a function from its conditional expecations

Question

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $\{ \mathcal{F}_i \}_{i \in \mathcal{I}}$ be a family of $\sigma$-subalgebras such that $$ \mathcal{F} = \sigma(\mathcal{F}_i, i \in \mathcal{I}). $$

Question

What are the conditions on $\mathcal{F}$ and $\{ \mathcal{F}_i \}_{i \in \mathcal{I}}$ such that, for all $f \in L^1$ with $E[f] = 0$, $$ f = \sum_i E[f|\mathcal{F}_i] \, a.s. ? $$

Total independence of $\{ \mathcal{F}_i\}$ would perhaps suffice. Are there more general assumptions available?

Answer 1

Whatever be the sigma algbebras the equation fails when $f=1$.