First-crossing maximal inequality for refining partitions ($\mu (\{{\sup }_k{\mathrm{avg}}_{M_k(x)}|f|>t\})\le \Vert f{\Vert }_1/t$)

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.

Assume in addition that the probability space is a metric space, continuous integrable functions are dense in $L^1$, and the diameters of its partition cells tend to zero at each point. For a continuous $g$, the oscillation average tends to zero because every point in $M_k(x)$ is within its diameter of $x$. For $h=f-g$, the upper limit of the oscillation of $f$ is bounded by the partition maximal function of $h$ plus $|h(x)|$. The first-crossing maximal inequality for refining partitions and Chebyshev's inequality bound the measure where this exceeds $2\epsilon$ by $2\Vert h{\Vert }_1/\epsilon$. Approximation proves the displayed almost everywhere assertion. Refinement supplies the maximal estimate; shrinking diameter supplies the approximation argument. Without shrinking diameter, a constant partition generally converges only to its cell averages.