We prove two results on approximation of Borel measures in the Kantorovich metric. The first result provides the rate of convergence in this metric of normalized surface measures on n -dimensional spheres to the standard Gaussian measure on the countable power of the real line, restricted to an arbitrary continuously embedded separable Banach space of full measure. The second result, for an arbitrary weakly convergent sequence of probability Borel measures on a separable Fréchet space, gives a sufficient condition for convergence in the Kantorovich metric generated by the norm of a compactly embedded separable Banach space.
V. I. Bogachev (Sun,) studied this question.