Hausdorff metric structure in the space of probability measures

The Hausdorrf distance between two bounded functions on the real line is defined by B. Sendov and B. Penkov. It is knows 2 that the Levy distance between distribution functions and the Hausdorff distance coinside. The connection of the structure of Levy–Prokhorov's distance with that of the Hausdorff distance is analysed in this paper.