 Dyadic occupancy approximation of a simple point measure

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.