Integral representation for Radon measures on arbitrary Hausdorf space

V. K. Zakharov ; A. V. Mikhalev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 4, pp. 1135-1172.

Résumé

After the fundamental articles of Riesz, Radon and Hausdorf in 1909–1914 the problem of general Radon representation became to be actual: for Hausdorf topological spaces to find a class of linear functionals isomorphic to the space of integrally represented Radon measures. Up to the beginning of 50-s a bijective solution of Radon representation for locally compact spaces was obtained by Halmos, Hewitt, Edwards, Bourbaki etc. For bounded Radon measures on a Tychonof space the problem of bijective representation was solved in 1956 by Yu. V. Prohorov. In 1975–1976 Topsoe and Pollard made an important step into consideration of the problem for an arbitrary Hausdorf topological space. On this way König in 1995–1997 has got a bijective version of Radon representation for isotone and positively-linear functionals on the cone of positive upper semicontinuous functions with compact support. In 1996–1997 the authors have got bijective and isomorphic versions of the general Radon representation. In this paper one of possible solution of the general Radon representation is exposed. For this reason the family of metasemicontinuous functions with compact support and the class of thin functionals are under consideration. Bijective and isomorphic versions of a solution (theorems 1 and 2 (II.5)) are given. To get an isomorphic version the family of Radon bimeasures is introduced.