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[EN] In this work we show how to define a probability measure with the help of a fractal structure. One of the keys of this approach is to use the completion of the fractal structure. Then we use the theory of a cumulative distribution function on a Polish ultrametric space and describe it in this context. Finally, with the help of fractal structures, we prove that a function satisfying the properties of a cumulative distribution function on a Polish ultrametric space is a cumulative distribution function with respect to some probability measure on the space., M. A. S´anchez-Granero acknowledges the support of grant MTM2015-64373-P (MINECO/FEDER, UE).

We give a new metrization theorem on terms of a new structure introduced by the authors in [Rend. Instit. Mat. Univ. Trieste 30 (1999) 21–30] and called fractal structure. This allows us to approach some classical and new metrization theorems (due to Nagata, Smirnov, Moore, Arhangel'skii, Frink, Borges, Hung, Morita, Fletcher, Lindgren, Williams, Collins, Roscoe, Reed, Rudin, Hanai, Stone, Burke, Engelking and Lutzer) from a new point of view.

In this paper we introduce a new structure on topological spaces which allows us to give a characterization of non-archimedeanly quasipseudometrizable spaces.