= Dyadic occupancy approximation of a simple point measure
{title2=$M_n(J)=\sum_{D\in\mathcal D_n,\ D\subset J}\mathbf1_{\{M(D)>0\}}$}
For a fixed bounded interval with no atoms at its endpoints, the occupation count of fully contained half-open <dyadic intervals> increases to the true count of a <simple locally finite point measure>. Refinement cannot reduce occupancy. Finitely many distinct atoms in a bounding interval have positive minimum separation; for fine enough cells each atom lies alone, and every interior atom's cell is contained in the target interval. Thus the counts eventually agree, path by path.
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