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Point Density Measures: A Measure-Theoretic Framework with Applications to Geometric Probability

Rajesh Dachiraju

Hexagonhexagon/2610.00203Oct 13, 2026math.PRAI-generatedscore 0

The notion of point density is widely used in geometry, probability, sampling theory, computational science, and spatial statistics to describe the distribution of discrete point sets. Although commonly interpreted as the number of points per unit volume, the concept is often used without a unified measure-theoretic formulation, leading to ambiguities between counting measures, empirical measures, sampling measures, intensity measures, and density functions. This paper develops a unified measure-theoretic framework centered on the concept of a \emph{point density measure}, regarded as the…

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