Geodesic
On soft mappings of the unit ball of Borel measures
Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 4, pp. 41-54.

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

The main result of this paper is two theorems. One of them asserts that the functor $U_\tau$ takes the 0-soft mappings between spaces of weight ${\leq}\,\omega_1$ and Polish spaces to soft mappings. The other theorem, which is a corollary to the first one, asserts that the functor $U_\tau$ takes the $\mathrm{AE}(0)$-spaces of weight ${\leq}\,\omega_1$ to $\mathrm{AE}$-spaces. These theorems are proved under Martin's axiom $\textup{MA}(\omega_1)$. The results cannot be extended to spaces of weight ${\geq}\,\omega_2$. For spaces of weight $\omega_1$, these results cannot be obtained without additional set-theoretic assumptions. Thus, the question as to whether the space $U_\tau(\mathbb R^{\omega_1})$ is an absolute extensor cannot be answered in ZFC. The main result cannot be transferred to the functor $U_R$ of the unit ball of Radon measures. Indeed, the space $U_R(\mathbb R^{\omega_1})$ is not real-compact and, therefore, $U_R(\mathbb R^{\omega_1})\notin\mathrm{AE}(0)$. 
@article{FPM_2003_9_4_a1,
     author = {Yu. V. Sadovnichii},
     title = {On soft mappings of the unit ball of {Borel} measures},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {41--54},
     publisher = {mathdoc},
     volume = {9},
     number = {4},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2003_9_4_a1/}
}
TY  - JOUR
AU  - Yu. V. Sadovnichii
TI  - On soft mappings of the unit ball of Borel measures
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2003
SP  - 41
EP  - 54
VL  - 9
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2003_9_4_a1/
LA  - ru
ID  - FPM_2003_9_4_a1
ER  - 
%0 Journal Article
%A Yu. V. Sadovnichii
%T On soft mappings of the unit ball of Borel measures
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2003
%P 41-54
%V 9
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2003_9_4_a1/
%G ru
%F FPM_2003_9_4_a1
Yu. V. Sadovnichii. On soft mappings of the unit ball of Borel measures. Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 4, pp. 41-54. http://geodesic.mathdoc.fr/item/FPM_2003_9_4_a1/

[1] Banakh T. O., Radul T. N., “Geometriya otobrazhenii prostranstv veroyatnostnykh mer”, Matem. stud. Pratsi Lvivskogo matem. t-va, 1999, no. 11, 17–30 | MR | Zbl

[2] Varadarain V. S., “Mery na topologicheskikh prostranstvakh”, Mat. sb., 55:1 (1961), 35–100

[3] Sadovnichii Yu. V., “O nekotorykh kategornykh svoistvakh funktora $U^{\tau}$”, Vestnik Mosk. un-ta. Ser. 1, Matematika, mekhanika, 1999, no. 3, 38–42 | MR

[4] Sadovnichii Yu. V., “Podnyatie funktorov $U_\tau$ i $U_R$ na kategoriyu ogranichennykh metricheskikh prostranstv i kategoriyu ravnomernykh prostranstv”, Mat. sb., 191:11 (2000), 79–104 | MR

[5] Sadovnichii Yu. V., “O svoistvakh polnoty funktorov edinichnogo shara borelevskikh mer”, Tr. seminara im. I. G. Petrovskogo, no. 23, 2003, 338–357 | MR

[6] Fedorchuk V. V., “Topologicheskaya polnota prostranstv mer”, Izv. RAN. Ser. mat., 63:4 (1999), 207–223 | MR | Zbl

[7] Fedorchuk V. V., Chigogidze A. Ch., Absolyutnye retrakty i beskonechnomernye mnogoobraziya, Nauka, M., 1992 | MR | Zbl

[8] Schepin E. V., “Funktory i neschetnye stepeni kompaktov”, Uspekhi mat. nauk, 36:3 (1981), 3–62 | MR | Zbl

[9] Ditor S., Haydon R., “On absolute retracts, $P(S)$ and complemented subspaces of $C(D^{\omega_1})$”, Studia Math., 56:3 (1976), 243–251 | MR | Zbl

[10] Gardner R. G., Pfeffer W. F., “Borel measures”, Handbook of set-theoretic topology, Elsevier, 1984, 961–1043 | MR