Preiss, D.

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Institute of Mathematics AS CR, v. v. i.

Toposym 3David PreissMetric spaces in which Prohorov's theorem is not validIn: Josef Novák (ed.): General Topology and its Relations to Modern Analysis and Algebra,Proceedings of the Third Prague Topological Symposium, 1971. Academia Publishing House of theCzechoslovak Academy of Sciences, Praha, 1972. pp. 359.Persistent URL: http://dml.cz/dmlcz/700750Terms of use:© Institute of Mathematics AS CR, 1972Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.cz359 METRIC SPACES IN WHICH PROHOROV'S THEOREM IS NOT VALID D. PREISS Praha A well-known Prohorov's theorem says that for every topologically complete metric space X9 for every compact set Af of measures on X with mass 1 (with the weak topology) and for every 6 > 0 there exists a compact set A a X such that fi(X — A) < < e for each /xeM. It has been a problem whether this theorem holds in every separable metric space which is a Borel subset of its completion. This problem can be solved by the help of the following theorem. Theorem 1. A coanalytic separable metric space is topologically complete if and only if it contains no countable dense-in-itself G6 subspace. If Prohorov's theorem holds in some metric space X then it is easy to prove that it holds also in every G5 subspace of X. With respect to Theorem 1 and to Sierpinski's result according to which every two countable dense-in-itself metric spaces are homeomorphic it is clear that the negative answer for some countable dense-in-itself metric space implies the negative answer for every separable coanalytic metric space which is not topologically complete. Such countable metric space can be constructed (namely the space of rational numbers), therefore the following theorem holds. Theorem 2.IfX is a separable coanalytic metric space then Prohorov's theorem holds in X if and only ifX is topologically complete. CHARLES UNIVERSITY, PRAHA 

Metric spaces in which Prohorov's theorem is not valid

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Metric spaces in which Prohorov's theorem is not valid

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Institute of Mathematics AS CR, v. v. i.