The Converse of Minlos' Theorem

Let ^ be the class of barrelled locally convex Hausdorff space E such that E'^ satisfies the property B in the sense of Pietsch. It is shown that if Ee^and if each continuous cylinder set measure on E' is <7(£', E) -Radon, then E is nuclear. There exists an example of non-nuclear Frechet space E such that each continuous Gaussian cylinder set measure on £'is 0(E', E)-Radon. Let q be 2 < q < oo. Suppose that E e ^ and £ is a projective limit of Banach space {Ea} such that the dual E'a is of cotype q for every ct,. Suppose also that each continuous Gaussian cylinder set measure on E' is a(E',E) -Radon. Then E is nuclear.