Geodesic
Some modifications of the Dudley metric
Zapiski Nauchnykh Seminarov POMI, Continuity and stability in the problems of probability theory and mathematical statistics, Tome 61 (1976), pp. 17-24 
N. V. Grigorevskii; I. S. Shiganov. Some modifications of the Dudley metric. Zapiski Nauchnykh Seminarov POMI, Continuity and stability in the problems of probability theory and mathematical statistics, Tome 61 (1976), pp. 17-24. http://geodesic.mathdoc.fr/item/ZNSL_1976_61_a1/
@article{ZNSL_1976_61_a1,
     author = {N. V. Grigorevskii and I. S. Shiganov},
     title = {Some modifications of the {Dudley} metric},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {17--24},
     year = {1976},
     volume = {61},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1976_61_a1/}
}
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%A I. S. Shiganov
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Upper and lower bounds are obtained for the modified Dudley metric in terms of the Levy–Prokhorov metric.