Lemma 3

Let $g$ be a continuous function from X into $\mathbf{C}$ and S its $\mu$-essential image. Suppose that S is not empty and that $0\notin S$; denote by $\delta$ the distance from 0 to S. Then $\delta >0$.

Put $h(x) = 1/g(x)$ if $g(x)\not = 0$ and $h(x) = 0$ otherwise. The function $h$ belongs to $\mathscr{L}^{\infty}(X, \mu)$. Let $\widetilde{h}$ be the class of $h$ in $L^{\infty}(X, \mu)$. Then we have the formula $\delta^{-1}=\|\widetilde{h}\|_{\infty}$.

Let U be an open neighbourhood of 0 such that the open set $Z =\overset{-1}{g}(U)$ is locally $\mu$-negligible, hence negligible (INT, IV, p. 172, § 5, n$^o2$, Cor. 2). Let Y be the support of $\mu$; we have $Y\subset X$ - Z. The restriction of the function $h$ to X - Z is continuous and bounded; therefore $h\in \mathscr{L}^{\infty}(X, \mu)$ and the norm of $\widetilde{h}$ in $\mathscr{L}^{\infty}(X, \mu)$ is equal to the norm of its restriction to X - Z. Moreover, for every $\alpha \in \mathbf{R}_+$, the set of $x\in X$ - Z such that $|h(x)|> \alpha$ is an open set in X - Z; therefore it is locally $\mu$-negligible if and only if it does not meet Y (INT, IV, loc. cit. and INT, III, p. 66, § 3, n$^o2$, Def. 1). Consequently, we have

$\|\widetilde{h}\|_{\infty}=$ sup$_{x\in Y}\frac{1}{|g(x)|}$,

whence

$\|\widetilde{h}1\|=$ inf$_{\infty x\in Y}|g(x)|=$ inf$_{\lambda\in g(Y)}|\lambda |=$ inf$_{\lambda\in\overline{g(Y)}}|\lambda |$

which is equal to $\delta$ by the preceding lemma.