Radial Integration in Continuous Dimension: A Mellin-Gamma Classification of Euclidean Ball Volume

TL;DR

This paper classifies positive linear functionals satisfying specific scaling and normalization conditions, showing they are uniquely represented by Mellin-Gamma measures, and derives the Euclidean ball volume formula as a consequence.

Contribution

It provides a rigidity proof that characterizes Mellin-Gamma measures solely based on axioms, without relying on analytic continuation or special functions.

Findings

Unique characterization of Mellin-Gamma measures under given axioms.

Derivation of Euclidean ball volume formula from measure mass.

Identification of dimension-shift structures via cocycles and a shifted Bohr--Mollerup theorem.

Abstract

We classify positive linear functionals on $C_c(\mathbb{R}_{>0})$ satisfying scaling covariance of degree $x/2$ and Gaussian normalization to $\pi^{x/2}$. We prove that the unique such functionals are represented by the Mellin--Gamma measures \[ d\mu_x(u) = \frac{\pi^{x/2}}{\Gamma(x/2)}\, u^{x/2 - 1}\, du, \quad x > 0. \] The result is a rigidity statement: the Mellin--Gamma structure is forced by the axioms, without assuming analytic continuation, special functions, or a priori formulas. The proof reduces the scaling condition, via a logarithmic change of variables, to translation invariance on $\mathbb{R}$, where Haar measure uniqueness determines the measure up to normalization, which is fixed by the Gaussian integral. As a consequence, the Euclidean ball volume formula \[ V(x) = \frac{\pi^{x/2}}{\Gamma(x/2 + 1)} \] is recovered as the mass of the unit interval. We further analyze the induced dimension-shift structure, identifying two multiplicative cocycles whose ratio is a coboundary given by the dimension function $x$, and give an independent characterization via a shifted Bohr--Mollerup theorem.