A counterexample to ``An extension of the Vitali-Hahn-Saks theorem'' and a compactness result

We give a counterexample to ``An extension of the Vitali-Hahn-Saks theorem'' and from that highlight the sharp frame within which any attempt to change the version of such an extension should be possible. Lastly a sequential compactness criterion for Radon measures absolutely continuous with respect to a prescribed Radon measure defined on a locally compact separable metric space (taking into account the ideas of Hernandez-Lerma and Lasserre) is proved. The results deal with Radon measures but yield obvious corollaries on real (or vector-valued) Radon measures and so on functions with bounded variation on open subsets of $\mathbf{R}^n$.