Exercise 16, § 6

Let $X, X'$ be two metrizable spaces. A mapping $f : X \to X'$ is said to be a Borel mapping if, for each closed subset $F'$ of $X'$, $\overline{f^{-1}(F')}$ is a Borel set in $X$. For each even (resp. odd) countable ordinal $\alpha$, $f$ is said to be of class $\alpha$ if, for each closed subset $F'$ of $X'$, $\overline{f^{-1}(F')}$ belongs to $\mathfrak{F}_\alpha(X)$ [resp. $\mathfrak{G}_\alpha(X)$] (Exercise 4); then, for each open set $G'$ in $X'$, $\overline{f^{-1}(G')}$ belongs to $\mathfrak{G}_\alpha(X)$ [resp. $\mathfrak{F}_\alpha(X)$]. The Borel mappings of class 0 are precisely the continuous mappings.

a) The characteristic function $\varphi_A$ of a subset $A$ of $X$ is of class $\alpha$ if and only if $A \in \mathfrak{F}_\alpha \cap \mathfrak{G}_\alpha$.

b) Let $f : X \to X'$ be a mapping of class $\alpha$. Show that for every subset $B' \in \mathfrak{F}_\beta(X')$ [resp. $B' \in \mathfrak{G}_\beta(X')$], $\overline{f^{-1}(B')}$ belongs to $\mathfrak{F}_{\alpha+\beta}(X)$ [resp. $\mathfrak{G}_{\alpha+\beta}(X)$]. (Use transfinite induction on $\beta$.)

c) Let $f : X \to X'$ be a mapping of class $\alpha$ and let $g : X' \to X''$ be a mapping of class $\beta$, where $X''$ is a third metrizable space. Show that $g \circ f$ is a Borel mapping of class $\alpha + \beta$.

d) For each even (resp. odd) countable ordinal $\alpha$, let $(A_n)$ be a sequence of subsets of $X$ which belong to $\mathfrak{G}_\alpha$ (resp. $\mathfrak{F}_\alpha$) and which cover $X$. If a mapping $f : X \to X'$ is such that $f|A_n$ is of class $\alpha$ for each $n$, then $f$ is of class $\alpha$. Give the analogous result for a finite sequence of subsets of $X$ which belong to $\mathfrak{F}_\alpha$ (resp. $\mathfrak{G}_\alpha$) and cover $X$.