Exercise 8, § 1

Let X, Y be two metric spaces and let $(f_n)$ be a sequence of Borel mappings of class $\alpha$ of X into Y (Chapter IX, § 6, Exercise 16).

a) Suppose that the sequence $(f_n)$ converges pointwise to a mapping $f : X \to Y$. Show that $f$ is of class $\alpha + 1$. [If U is an open subset of Y, note that $\bar{f}^1(U) = \bigcup_{n \geq 1} \left( \bigcap_{k \geq 0} \bar{f}_{n+k}^1(U) \right)$, and use Chapter IX, § 6, Exercise 4.]

b) Suppose that the sequence $(f_n)$ converges uniformly to $f$. Show that $f$ is of class $\alpha$. [Let F be a closed subset of Y, and for each integer $n > 0$, let $V_n$ be the set of points of Y whose distance from F is $\leq 1/n$. Show that there is an increasing sequence $n \to m(n)$ of integers such that $\bar{f}^1(F) = \bigcap_n \bar{f}_{m(n)}^1(V_n)$.]

c) Suppose that Y is of countable type. Show that, if $f : X \to Y$ is a Borel function of class $\alpha > 0$, there exists a sequence of Borel functions $g_n : X \to Y$, of class $< \alpha$, such that $(g_n)$ converges pointwise to $f$. [Show that we may take the $g_n$ to be functions which take only a finite number of values; use Exercises 4 h) and 16 c) of Chapter IX, § 6.]