Positive linear functionals on C_c(R>0) with scaling covariance of degree x/2 and Gaussian normalization are uniquely the Mellin-Gamma measures, recovering V(x) = π^{x/2}/Γ(x/2 + 1) as the mass of the unit interval.
Deligne, La catégorie des représentations du groupe symétriqueSt, lorsquetn’est pas un entier naturel, inAlgebraic Groups and Homogeneous Spaces, Tata Inst
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Radial Integration in Continuous Dimension: A Mellin-Gamma Classification of Euclidean Ball Volume
Positive linear functionals on C_c(R>0) with scaling covariance of degree x/2 and Gaussian normalization are uniquely the Mellin-Gamma measures, recovering V(x) = π^{x/2}/Γ(x/2 + 1) as the mass of the unit interval.