Geodesic
The Riesz–Radon Problem of Characterizing Integrals and the Weak Compactness of Radon Measures
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 106-116 
V. K. Zakharov. The Riesz–Radon Problem of Characterizing Integrals and the Weak Compactness of Radon Measures. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 106-116. http://geodesic.mathdoc.fr/item/TM_2005_248_a10/
@article{TM_2005_248_a10,
     author = {V. K. Zakharov},
     title = {The {Riesz{\textendash}Radon} {Problem} of {Characterizing} {Integrals} and the {Weak} {Compactness} of {Radon} {Measures}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {106--116},
     year = {2005},
     volume = {248},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2005_248_a10/}
}
TY  - JOUR
AU  - V. K. Zakharov
TI  - The Riesz–Radon Problem of Characterizing Integrals and the Weak Compactness of Radon Measures
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2005
SP  - 106
EP  - 116
VL  - 248
UR  - http://geodesic.mathdoc.fr/item/TM_2005_248_a10/
LA  - ru
ID  - TM_2005_248_a10
ER  - 
%0 Journal Article
%A V. K. Zakharov
%T The Riesz–Radon Problem of Characterizing Integrals and the Weak Compactness of Radon Measures
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2005
%P 106-116
%V 248
%U http://geodesic.mathdoc.fr/item/TM_2005_248_a10/
%G ru
%F TM_2005_248_a10

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The problem of characterizing integrals considered in this paper dates back to the fundamental works of Riesz (1909), Radon (1913), and Frechet 1914). A solution to this problem is given in the form of a general parametric theorem, which implies the following theorems as particular cases: (1) the Riesz–Radon theorem for a locally compact space, (2) the Prokhorov theorem for a Tikhonov space, and (3) an integral representation theorem for an arbitrary Hausdorff space. A weak compactness criterion for the sets of bounded Radon measures on an arbitrary Hausdorff space is derived as an application of the last theorem. This criterion dates back to the Prokhorov criterion for a Polish space and to the Prokhorov–Le Cam theorem for a Tikhonov space.

[1] Fremlin D.H., Topological Riesz spaces and measure theory, Cambridge Univ. Press, Cambridge, 1974 | MR

[2] Hewitt E., Stromberg K., Real and abstract analysis, Springer, Berlin, 1965 | MR

[3] Riesz F., “Sur les opérations fonctionelles linéaires”, C. r. Acad. sci. Paris, 149 (1909), 974–977

[4] Radon J., “Theorie und Anwendungen der absolut additiven Mengenfunktionen”, Sitzungsber. Akad. Wiss. Wien., 122 (1913), 1295–1438 | Zbl

[5] Halmos P.R., Measure theory, Van Nostrand, Princeton, 1950 | MR | Zbl

[6] Hewitt E., “Integration on locally compact spaces. I”, Univ. Washington Publ. Math., 3 (1952), 71–75 | MR

[7] Edwards R.E., “A theory of Radon measures on locally compact spaces”, Acta math., 89 (1953), 133–164 | DOI | MR | Zbl

[8] Prokhorov Yu.V., “Skhodimost sluchainykh protsessov i predelnye teoremy teorii veroyatnostei”, Teor. veroyatn. i ee prim., 1 (1956), 177–238 | MR | Zbl

[9] Zakharov V.K., Mikhalev A.V., “Problema Radona dlya regulyarnykh mer na proizvolnom khausdorfovom prostranstve”, Fund. i prikl. matematika, 3:3 (1997), 801–808 | MR | Zbl

[10] Zakharov V.K., Mikhalev A.V., “Integralnoe predstavlenie dlya radonovskikh mer na proizvolnom khausdorfovom prostranstve”, Fund. i prikl. matematika, 3:4 (1997), 1135–1172 | MR | Zbl

[11] Zakharov V.K., Mikhalev A.V., “Problema integralnogo predstavleniya dlya radonovskikh mer na proizvolnom khausdorfovom prostranstve”, Dokl. RAN, 360:1 (1998), 13–15 | MR | Zbl

[12] Zakharov V.K., Mikhalev A.V., “Problema obschego radonovskogo predstavleniya dlya proizvolnogo khausdorfova prostranstva”, Izv. RAN. Ser. mat., 63:5 (1999), 37–82 | MR | Zbl

[13] Zakharov V.K., Mikhalev A.V., “Svyaz mezhdu integralnymi radonovskimi predstavleniyami dlya lokalno kompaktnogo i khausdorfova prostranstv”, Fund. i prikl. matematika, 7:1 (2001), 33–46 | MR | Zbl

[14] Zakharov V.K., Mikhalev A.V., “Problema obschego radonovskogo predstavleniya dlya proizvolnogo khausdorfova prostranstva. II”, Izv. RAN. Ser. mat., 66:6 (2002), 3–18 | MR

[15] König H., Measure and integration, Springer, Berlin, 1997 | MR

[16] Zakharov V.K., “Problema kharakterizatsii radonovskikh integralov”, Dokl. RAN, 385:6 (2002), 735–737 | MR

[17] Jacobs K., Measure and integral, Acad. Press, New York, 1978 | MR

[18] Bogachev V.I., Osnovy teorii mery, t. 1, 2, R Dynamics, Moskva; Izhevsk, 2003

[19] Vulikh B.Z., Vvedenie v teoriyu poluuporyadochennykh prostranstv, Fizmatgiz, M., 1961 | MR