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 $x\mathrm{/}2$ and Gaussian normalization to ${\pi }^{x\mathrm{/}2}$. We prove that the unique such functionals are represented by the Mellin–Gamma measures $$d{\mu }_x(u)=\frac{{\pi }^{x\mathrm{/}2}}{\mathrm{\Gamma }(x\mathrm{/}2)}{u}^{x\mathrm{/}2-1}du,x>0.$$ 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 $$V(x)=\frac{{\pi }^{x\mathrm{/}2}}{\mathrm{\Gamma }(x\mathrm{/}2+1)}$$ 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.