Geodesic
Continuous Selections as Parametrically Defined Integrals
Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 2, pp. 89-94.

Voir la notice de l'article provenant de la source Math-Net.Ru

An analog of the classical Michael theorem on continuous single-valued selections of lower semicontinuous maps whose values are closed and convex in a Fréchet space is proved for maps into metrizable (non-locally-convex) vector spaces. It turns out that, instead of the local convexity of the whole space containing these values, it is sufficient to require that the family of values of the map be \del{pointwise} uniformly locally convex. In contrast to the standard selection theorems, the proof bypasses the process of successively improving the approximations, and the desired selection is constructed as the result of pointwise integration with respect to a suitable probability distribution.
Keywords: continuous selection, convex-valued map, non-locally-convex vector space, probability measure.
Mots-clés : paracompact space 
@article{FAA_2008_42_2_a12,
     author = {P. V. Semenov},
     title = {Continuous {Selections} as {Parametrically} {Defined} {Integrals}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {89--94},
     publisher = {mathdoc},
     volume = {42},
     number = {2},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a12/}
}
TY  - JOUR
AU  - P. V. Semenov
TI  - Continuous Selections as Parametrically Defined Integrals
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2008
SP  - 89
EP  - 94
VL  - 42
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a12/
LA  - ru
ID  - FAA_2008_42_2_a12
ER  - 
%0 Journal Article
%A P. V. Semenov
%T Continuous Selections as Parametrically Defined Integrals
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2008
%P 89-94
%V 42
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a12/
%G ru
%F FAA_2008_42_2_a12
P. V. Semenov. Continuous Selections as Parametrically Defined Integrals. Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 2, pp. 89-94. http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a12/

[1] P. S. Aleksandrov, B. A. Pasynkov, Vvedenie v teoriyu razmernosti, Nauka, Moskva, 1973 | MR

[2] U. Rudin, Funktsionalnyi analiz, Mir, Moskva, 1975 | MR

[3] P. V. Semenov, E. V. Schepin, Funkts. analiz i ego pril., 26:2 (1992), 36–40 | MR | Zbl

[4] C. Bessaga, A. Pelczynski, Selected Topics in Infinite-Dimensional Topology, PWN – Polish Scientific Publishers, Warsaw, 1975 | MR | Zbl

[5] T. Dobrowolski, Proc. Amer. Math. Soc., 93:3 (1985), 555–560 | DOI | MR | Zbl

[6] T. Dobrowolski, J. van Mill, Fund. Math., 192:3 (2006), 215–232 | DOI | MR | Zbl

[7] O. Hadžić, Lecture Notes in Math., 948, Springer-Verlag, Berlin–New York, 1981, 118–130 | DOI | MR

[8] V. Klee, Math. Ann., 141 (1960), 286–296 | DOI | MR | Zbl

[9] E. Michael, Canad. J. Math., 11 (1959), 556–575 | DOI | MR | Zbl

[10] E. Michael, Proc. Amer. Math. Soc., 17 (1966), 1404–1406 | DOI | MR | Zbl

[11] S. Park, Topology Appl., 135:1–3 (2004), 197–206 | DOI | MR | Zbl

[12] D. Repovš, P. V. Semenov, E. V. Ščepin, Topology Appl., 54:1–3 (1993), 77–83 | DOI | MR | Zbl

[13] J. Roberts, Canad. J. Math., 30:3 (1978), 449–454 | DOI | MR | Zbl