= First-crossing maximal inequality for refining partitions
{title2=$\mu(\{\sup_k\operatorname{avg}_{M_k(x)}|f|>t\})\leq\|f\|_1/t$}
Let a <probability space> have countable measurable partitions of positive-<measure> cells, each partition refining its predecessor. An exceeding point has a first level where its cell average of $|f|$ exceeds $t$. The cells at those first crossings are disjoint: intersecting cells at different levels are nested, and an earlier exceeding cell prevents any of its descendants from being a first crossing. They cover exactly the maximal superlevel set. Summing their strict average inequalities gives the displayed <weak type (1,1)> estimate with constant one. The same proof works on a general <measure> space when every relevant cell has positive finite <measure>.
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