Proposition 4

Let $g\in \mathscr{L}^{\infty}(X, \mu)$. Denote by $\widetilde{g}$ the class of $g$ in $L^{\infty}(X, \mu)$ and by S the spectrum of $\widetilde{g}$. The mapping $f\mapsto f\circ g$ (remark, p. 184) is the functional calculus mapping of $\mathscr{C}(S)$ into $L^{\infty}(X, \mu)$ associated with $\widetilde{g}$.

Let $f\in \mathscr{C}(S)$. The function $h=f\circ g$ is $\mu$-measurable (Lemma 5 of IV, p. 184) and bounded. Denote by $f\widetilde{\circ}g$ its class in $L^{\infty}(X, \mu)$. The mapping $f\mapsto f\widetilde{\circ}g$ is a continuous morphism of involutive algebras with unit of $\mathscr{C}(S)$ into $L^{\infty}(X, \mu)$. Since $g(x)$ belongs to S locally $\mu$-almost everywhere (Lemma 1 of IV, p. 181 and Lemma 6), this morphism associates with the identity mapping of S the class $\widetilde{g}$ of the function $g$. It therefore coincides with the functional calculus mapping of $\widetilde{g}$ (Prop. 7 of I, p. 111).