  Assume that we are given a filtration $(\mathscr F_n)$ on a probability space
$(\Omega,\mathscr F,\mathbb P)$ of the form that each $\mathscr F_n$ is
generated by the partition of one atom of $\mathscr F_{n-1}$ into two atoms of
$\mathscr F_n$ having positive measure. Additionally, assume that we are given
a finite-dimensional linear space $S$ of $\mathscr F$-measurable, bounded
functions on $\Omega$ so that on each atom $A$ of any $\sigma$-algebra
$\mathscr F_n$, all $L^p$-norms of functions in $S$ are comparable
independently of $n$ or $A$. Denote by $S_n$ the space of functions that are
given locally, on atoms of $\mathscr F_n$, by functions in $S$ and by $P_n$ the
orthoprojector (with respect to the inner product in $L^2(\Omega)$) onto $S_n$.
Since $S = \operatorname{span}\{1_\Omega\}$ satisfies the above assumption and
$P_n$ is then the conditional expectation $\mathbb E_n$ with respect to
$\mathscr F_n$, for such filtrations, martingales $(\mathbb E_n f)$ are special
cases of our setting. We show in this article that certain convergence results
that are known for martingales (or rather martingale differences) are also true
in the general framework described above. More precisely, we show that the
differences $(P_n - P_{n-1})f$ converge unconditionally and are democratic in
$L^p$ for $1<p<\infty$. This implies that those differences form a greedy basis
in $L^p$-spaces for $1<p<\infty$.
