Dyadic filtration (${\mathcal{F}}_n=\sigma \{\text{dyadic cubes of side }{2}^{-n}\}$)

On $(0,1{]}^d$, the dyadic filtration is generated by the partitions into half-open cubes of side $2^{-n}$. The conditional expectation of an integrable function is its average on each atom. The increasing partitions generate the Borel sigma-algebra. Step functions on these partitions are dense in $L^1$; approximation and the Doob maximal inequality show that their conditional averages converge in $L^1$ and almost everywhere to the original function.