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

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.