= Dyadic filtration
{title2=$\mathcal F_n=\sigma\{\text{dyadic cubes of side }2^{-n}\}$}
On $(0,1]^d$, the dyadic <filtration (probability theory)> 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.
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