= Dyadic filtration
{title2=$\mathcal F_n=\sigma\{\text{dyadic cubes of side }2^{-n}\}$}
On $(0,1]^d$ with <Lebesgue measure>[Lebesgue probability measure], 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 of a measure>[atom]. The increasing partitions generate the <Borel sigma-algebra>. <step function>[Step functions] on these partitions are <dense subset>[dense] in $L^1$; approximation and the <Doob maximal inequality> show that their <conditional expectation>[conditional averages] converge in $L^1$ and <almost everywhere> to the original <function>.
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