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 Fundam. Prikl. Mat.: Year: Volume: Issue: Page: Find

 Fundam. Prikl. Mat., 2003, Volume 9, Issue 4, Pages 41–54 (Mi fpm749)

On soft mappings of the unit ball of Borel measures

M. V. Lomonosov Moscow State University

Abstract: 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 $MA(\omega_1)$. 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, $U_R(\mathbb R^{\omega_1})\notin\mathrm{AE}(0)$.

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English version:
Journal of Mathematical Sciences (New York), 2006, 136:5, 4156–4165

Bibliographic databases:

UDC: 515.12

Citation: Yu. V. Sadovnichii, “On soft mappings of the unit ball of Borel measures”, Fundam. Prikl. Mat., 9:4 (2003), 41–54; J. Math. Sci., 136:5 (2006), 4156–4165

Citation in format AMSBIB
\Bibitem{Sad03} \by Yu.~V.~Sadovnichii \paper On soft mappings of the unit ball of Borel measures \jour Fundam. Prikl. Mat. \yr 2003 \vol 9 \issue 4 \pages 41--54 \mathnet{http://mi.mathnet.ru/fpm749} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2093412} \zmath{https://zbmath.org/?q=an:1073.54007} \elib{http://elibrary.ru/item.asp?id=9068286} \transl \jour J. Math. Sci. \yr 2006 \vol 136 \issue 5 \pages 4156--4165 \crossref{https://doi.org/10.1007/s10958-006-0225-4} \elib{http://elibrary.ru/item.asp?id=13506292} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33745665853}