A continuity property of the convolution

It is shown that Pn --> P0 weakly (i.e. [latin small letter f with hook][latin small letter f with hook]dPn --> [latin small letter f with hook][latin small letter f with hook]dP0 for every continuous and bounded function [latin small letter f with hook] defined on ) for probability measures Pn, n[epsilon][union or logical sum]{0}, on the Borel subssets of , implies in the sense of Brooks and Chacon for any probability measure P on the Borel subsets of .