Exercise 10, § 1

Let T be a topological space and $\mu$ a positive measure on T; assume that T is the union of a sequence of $\mu$-integrable sets $T_n$ ($n \geq 0$). Denote by $\mathfrak{C}$ the clan of subsets of T generated by the compact subsets. a) Let X be a Lusin space. For every $\mu$-measurable mapping $f$ of T into X, there exists a sequence of mappings $f_m : T \to X$ having the following properties: $\alpha)$ for $\mu$-almost every $t \in T$, $f(t) = \lim_{m \to \infty} f_m(t)$; $\beta)$ for every integer $m$, there exists a finite partition of T into sets $A_j \in \mathfrak{C}$ such that $f_m$ is constant on each of the sets $A_j$ (reduce to the case that the sets $T_n$ form a partition of T, where $T_0$ is $\mu$-negligible and $T_n$ is compact for $n \geq 1$, and where X is a Polish space). b) Let $X_1, \ldots, X_p$ be Lusin spaces, Y a topological space, $g$ a mapping of $X_1 \times \cdots \times X_p$ into Y, and $f_i$ (for $1 \leq i \leq p$) a $\mu$-measurable mapping of T into $X_i$. Assume that, for $1 \leq i \leq p$, the partial mapping $x_i \mapsto g(x_1, \ldots, x_p)$ of $X_i$ into Y is continuous for any $x_1, \ldots, x_{i-1}, x_{i+1}, \ldots, x_p$. Show that the mapping $t \mapsto g(f_1(t), \ldots, f_p(t))$ of T into Y is $\mu$-measurable.