Corollary 3

Let $u$ be a partial normal operator on a Hilbert space E. Let $f\in \mathscr{L}_u$(Sp($u$)). For all $x$ and $y$ in E, the spectral measure of $(x, y)$ relative to $f(u)$ is the image measure $f(\mu)$, where $\mu$ is the spectral measure of $(x, y)$ relative to $u$.

In view of the spectral theorem (th. 1 of IV, p. 266), one may suppose that $u$ is the multiplication operator $m_g$ on $L^2(X, \mu)$, where X is a locally compact topological space, $\mu$ a positive measure on X and $g\in \mathscr{C}(X)$. Let $f_1$ and $f_2$ be in $\mathscr{L}^2(X, \mu)$, with classes $\widetilde{f}_1$ and $\widetilde{f}_2$ in $L^2(X, \mu)$. Since $f(m_g) =m_{f\circ g}$ (prop. 4, c)), the spectral measure of $(\widetilde{f}_1,\widetilde{f}_2)$ relative to $f(m_g)$ is the image measure $(f\circ g)(\overline{f}_1f_2\cdot \mu)$ (lemma 4, b) of IV, p. 269). This measure is equal to the image measure $f(g(f_1f_2\cdot \mu))$ (INT, V, p. 72, § 6, n$^o4$, prop. 4, a)), whence the assertion (lemma 4, b) of IV, p. 269).