Indistinguishability of absolutely continuous and singular distributions

It is shown that there are no consistent decision rules for the hypothesis testing problem of distinguishing between absolutely continuous and purely singular probability distributions on the real line. In fact, there are no consistent decision rules for distinguishing between absolutely continuous distributions and distributions supported by Borel sets of Hausdorff dimension 0. It follows that there is no consistent sequence of estimators of the Hausdorff dimension of a probability distribution