The Minlos lemma for positive-definite functions on additive subgroups of ${\displaystyle {ℝ}^n}$

Let H be a real Hilbert space. It is well known that a positive-definite function φ on H is the Fourier transform of a Radon measure on the dual space if (and only if) φ is continuous in the Sazonov topology (resp. the Gross topology) on H. Let G be an additive subgroup of H and let ${\displaystyle G_{pc}^{\wedge }}$ (resp. ${\displaystyle G_b^{\wedge }}$) be the character group endowed with the topology of uniform convergence on precompact (resp. bounded) subsets of G. It is proved that if a positive-definite function φ on G is continuous in the Gross topology, then φ is the Fourier transform of a Radon measure μ on ${\displaystyle G_{pc}^{\wedge }}$; if φ is continuous in the Sazonov topology, μ can be extended to a Radon measure on ${\displaystyle G_b^{\wedge }}$.