Proposition 11

Let $X$ be a metrizable space and let $f$ be a lower semi-continuous function on $X$ which takes its values in a closed interval $[a, b]$ of $\overline{\mathbf{R}}$. Then $f$ is the upper envelope of an increasing sequence of continuous functions on $X$ which take their values in $[a, b]$.

By transfer of structure we may assume that $a = 0$ and $b = 1$.

(i) Suppose first that $f = \varphi_A$, where $A$ is an open subset of $X$. Then the function $g_n$ defined by $g_n(x) = n \inf (d(x, X - A), 1/n)$ is continuous and $\geq 0$ on $X$; also $g_n(x) = f(x)$ when $x \in X - A$ and when

It follows immediately that $f = \sup_n (g_n)$.

(ii) In the general case consider, for each integer $n \geq 1$, the finite decreasing sequence of open sets

the function $g_n = \frac{1}{n} \sum_{k=1}^{n-1} \varphi_{A_k}$ is lower semi-continuous, and we have $0 \leq f(x) - g_n(x) \leq 1/n$ for all $n$; hence $f$ is the upper envelope of the sequence $(g_n)$. On the other hand, $g_n$ is a linear combination with positive coefficients of a finite number of characteristic functions of open sets and is therefore the upper envelope of a countable sequence $(h_{mn})_{m \geq 0}$ of continuous functions $\geq 0$, by (i); hence $f = \sup_{m,n} h_{mn}$. If we put we see that the sequence $(f_n)$ is an increasing sequence of continuous functions $\geq 0$, with $f$ as upper envelope, and which do not take the value 1, since $g_n \leq n - 1/n$.