Absolute $\sigma$-retracts and Luzin's theorem

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 $AR_\sigma(\mathfrak{M})$.