The Bochner-Minlos theorem

If X is a topological space, and for m ≥ n the maps πm,n : X → X are defined by (πm,n(x))(j) = x(j), j ∈ {1, . . . , n}, then the spaces X and maps πm,n constitute a projective system, and its limit in the category of topological spaces is XN with the maps πn : X N → X, where XN has the initial topology for the family {πn : n ∈ N} (namely, the product topology). We say that a function f : XN → R depends on only finitely many coordinates if there is some n and some function g : X → R such that f = g ◦ πn. We denote by Cfin(X) the set of all continuous functions XN → R that depend on only finitely many coordinates. If (X, τX) is a noncompact locally compact Hausdorff space, write Ẋ = X ∪ {∞}, and let τ be the collection of all subsets U of Ẋ such that either (i) U ∈ τX or (ii) ∞ ∈ U and X \ U is compact in (X, τX). One proves that (Ẋ, τ) is a compact Hausdorff space and that the inclusion map ι : X → Ẋ is a homeomorphism X → ι(X), where ι(X) has the subspace topology inherited from Ẋ. Also, if f ∈ C(X) then there is some F ∈ C(Ẋ) whose restriction to X equals f if and only if there is some g ∈ C0(X) and some constant c such