Dyadic occupancy approximation of a simple point measure ($M_n(J)={\sum }_{D\in {\mathcal{D}}_n,D\subset J}{1}_{\{M(D)>0\}}$)

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.