Simple locally finite point measure ($M={\sum }_j{\delta }_{x_j}$)

A point measure is a countable sum of Dirac measures. It is simple if all its atoms have mass one, and locally finite if every bounded set has finite mass. Simplicity lets fine enough cells count individual points, whereas occupation indicators would lose multiplicities in a nonsimple 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.