Isotropic probability measures in infinite-dimensional spaces

Every isotropic probability measure on the space R ^∞ of real sequences x = ( x ₁ , x ₂ ,...) is a convex combination of the measure concentrated at 0 and a member of I ₀ ( R ^∞ ), the set of all isotropic probability measures p _∞ on R ^∞ with p _∞ ({0}) = 0. Each p _∞ [unk] I ₀ ( R ^∞ ) is completely determined by any one of its finite-dimensional marginal distributions p _n . Each p _n has a density function f _n with dp _n ( x ₁ ,..., x _n ) = dx ₁ ... dx _n f _n ( x ₁ ² +... + x _n ² ). Each f _n is completely monotone in 0 < ξ < ∞ (hence analytic in the right complex ξ half-plane), and π ^{n /2} Γ( n /2) ^-1 ʃ ₀ ^∞ d ξ ξ ^{n /2-1} f _n (ξ) = 1. Every f that satisfies these two conditions is f _n for a unique p _∞ [unk] I ₀ ( R ^∞ ). Hence the equation πʃ _ξ ^∞ d ζ f ₂ (ζ) = ʃ ₀ ^∞ d μ ( t ) e ^{- t ξ} defines a bijection between I ₀ ( R ^∞ ) and the set of all probability measures μ on 0 ≤ t < ∞. If p _∞ [unk] I ₀ ( R ^∞ ) then p _∞ ({x: Σ _{i =1} ^∞ x _i ² < ∞}) = 0, so p _∞ is not a “softened” or “fuzzy” version of the inequality Σ _{i =1} ^∞ x _i ² ≤ 1. If the prior information in a linear inverse problem consists of this inequality and nothing else, stochastic inversion and Bayesian inference are both unsuitable inversion techniques.