THE CONVERSE OF MINLOS THEOREM

Let M be the class of barrelled locally convex Hausdorff space E such that E(b)' satisfies the property B in the sense of Pietsch. It is shown that if E epsilon M and if each continuous cylinder set measure on E' is sigma(E' 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 E' is sigma(E', E)-Radon. Let q be 2 less than or equal to q < infinity. Suppose that E epsilon M and E is a projective limit of Banach space {E(alpha)} such that the dual E(alpha)' is of cotype q for every alpha. Suppose also that each continuous Gaussian cylinder set measure on E' is sigma(E', E)-Radon. Then E is nuclear.