Absolute $\sigma$-retracts and Luzin's theorem
Matematičeskie zametki SVFU, Tome 25 (2018) no. 2, pp. 55-64
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We prove some properties of absolute $\sigma$-retracts. The generalization of the classical Luzin theorem about approximation of a measurable mapping by continuous mappings is given. Namely, we prove the following statement: Theorem. Let $Y$ be a complete separable metric space in $AR_\sigma(\mathfrak M)$, where $AR_\sigma(\mathfrak M)$ is the whole complex of all absolute $\sigma$-retracts. Suppose that $X$ is a normal space, $A$ is a closed subset in $X$, $\mu\geq0$ is the Radon measure on $A$, and $f\colon A\to Y$ is a $\mu$-measurable mapping. Given $\varepsilon>0$, there exist a closed subset $A_\varepsilon$ of $A$ such that $\mu(A\setminus A_\varepsilon)\leq\varepsilon$ and a continuous mapping $f_\varepsilon\colon X\to Y$ such that $f_\varepsilon(x)=f(x)$ for all $x\in A_\varepsilon$. Note that a connected separable $ANR(\mathfrak{M})$-space belongs to $AR_\sigma(\mathfrak{M})$.
Keywords:
absolute $\sigma$-retract, Luzin's theorem.
@article{SVFU_2018_25_2_a5,
author = {P. V. Chernikov},
title = {Absolute $\sigma$-retracts and {Luzin's} theorem},
journal = {Matemati\v{c}eskie zametki SVFU},
pages = {55--64},
year = {2018},
volume = {25},
number = {2},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/SVFU_2018_25_2_a5/}
}
P. V. Chernikov. Absolute $\sigma$-retracts and Luzin's theorem. Matematičeskie zametki SVFU, Tome 25 (2018) no. 2, pp. 55-64. http://geodesic.mathdoc.fr/item/SVFU_2018_25_2_a5/