= Solution
Every $x$ in the maximal superlevel set has a first index $k(x)$ at which the average on its unique cell $M_k(x)\in\mathcal P_k$ exceeds $t$. Refinement implies that all earlier cells containing $x$ contain this cell, and their averages are at most $t$. Thus $M_{k(x)}(x)\in\mathcal S_t$ and the selected cells cover the superlevel set.
Two cells at different levels are either disjoint or nested, because of refinement and the partition property. If selected cells of levels $k<l$ intersect, the level-$l$ one lies in the earlier cell whose average is above $t$, contradicting its first-crossing property. Cells at the same level are disjoint unless identical. Hence \b[the selected cells form a countable disjoint partition of the superlevel set]. A cell repeated unchanged at successive levels is counted once, at its first crossing; equal earlier ancestors are included in the comparison condition.
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