Entropy-visible multiscale metrics for L 1 functions

Abstract Measuring the difference between integrable functions or probability distributions is central to analysis, information theory, and physics. Classical distances and divergences provide global measures of discrepancy, but do not indicate how differences are distributed across intrinsic structural or informational resolutions. As a result, distributions that differ substantially in their internal organization may appear indistinguishable under standard metrics. This paper develops an entropy-visible multiscale representation for comparing nonnegative L 1 functions. Building on an entropy-driven hierarchical decomposition, each function is represented by a canonical sequence of step-function approximations in which refinement occurs only when genuine entropy-visible nonuniformity is present. Using these representations, we define finite-depth and infinite-depth pseudo-metrics that compare functions by resolving and accumulating discrepancies as entropy-defined distinctions become visible. Unlike single-scale distances, the proposed metric distinguishes purely coarse differences from structural disagreements that persist across resolutions. The representation further admits a canonical localization by scale and region, yielding an explicit description of where and by how much two functions differ in an information-theoretic sense. This provides a principled, entropy-aware notion of distance suited to coarse-graining and multiscale analysis in physical systems.