Proposition 9

— If F is a metrizable compact space, then every measurable mapping f of X into F is the uniform limit (on all of X) of a sequence of measurable step functions.

Let $d$ be a metric compatible with the topology of F. For every positive integer $n$ there exists a finite number of points $a_k \in \mathrm{F}$ such that the closed balls $B_k$ with center $a_k$ and radius $1/n$ form a covering of F; the sets $A_k = f^{-1}(B_k)$ are therefore measurable (Prop. 7) and form a covering of X. Consequently (§ 4, No. 9, Lemma) there exists a partition $(C_i)$ of X into a finite number of measurable sets such that each $A_k$ is the union of certain of the $C_i$. Let $c_i$ be a point of $C_i$ and let $g_n$ be the measurable step function equal to $f(c_i)$ on $C_i$ (for each index $i$). It is clear that $d(f(x), g_n(x)) \leqslant 2/n$ for all $x \in \mathrm{X}$.