Lemma 4

Let X be a locally compact topological space, let $\mu$ be a positive measure on X, and let $g$ be a $\mu$-measurable function on X. Let $m_g$ be the operator of multiplication by $g$ on $L^2(X, \mu)$.

a) The mapping $f\mapsto f(m_g)$ of $\mathscr{K}$ (Sp($m_g$)) into $\mathscr{L}(L^2(X, \mu))$ is given by $f\mapsto m_{f\circ g}$;

b) For $f_1$ and $f_2$ in $\mathscr{L}^2(X, \mu)$ of classes $\widetilde{f}_1$ and $\widetilde{f}_2$ in $L^2(X, \mu)$, the spectral measure dede of the image measure $g(\frac{(}{f}\widetilde{f}_{11}f,_2\widetilde{f}\cdot^2\mu)$ )relative to. $m_g$ is the restriction to Sp($m_g$)

The partial operator $m_g$ is normal and $b(m_g) =m_{\beta\circ g}=m_{g(1+|g|^2)^{-1/2}}$ (lemma 2). Moreover $\beta (g(x))$ belongs to Sp($m_{\beta\circ g}$) for every $x$ outside a locally $\mu$-negligible set $Y\subset X$ (proposition 22 of IV, p. 252 and lemma 1 of IV, p. 181). Since $f_{\beta}(\beta (g(x))) =f(g(x))$ for every $x\in X$ - Y, it follows that $f(m_g) =m_{f\circ g}$ from definition 3 and the cor. of prop. 6 of IV, p. 187.

Let $f_1$ and $f_2$ be in $\mathscr{L}^2(X, \mu)$ of classes $\widetilde{f}_1$ and $\widetilde{f}_2$ in $L^2(X, \mu)$. Since the measure $\nu =f_1f_2\cdot \mu$ is bounded, the image measure $g(\nu )$ is defined (INT, V, p. 69, § 6, n$^o1$, rem. 1). Let $f\in \mathscr{K}$ (Sp($m_g$)). We have $f(m_g) =m_{f\circ g}$ by a), whence

(INT, V, p. 69, § 6, n$^o1$, formula (1)), which establishes assertion b).