Absolute σ-retracts and Luzin’s theorem

Abstract

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 $f:A\rightarrow Y$ is a $\mu$-measurable mapping. Given $\varepsilon>0$, there exist a closed subset $A_{\varepsilon}$ of $A$ such that $\mu(A\backslash A_{\varepsilon})\leq\varepsilon$ and a continuous mapping $f_{\varepsilon}:X\rightarrow Y$ such that $f_{\varepsilon}(x)=f(x)$ for all $x\in A_{\varepsilon}$.
Note that a connected separable $ANR(\mathfrak{M})$-space belongs to $AR_{\sigma}(\mathfrak{M})$.