 First-crossing maximal inequality for refining partitions

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.