Corollary

Let $f$ and $g$ be in $\mathscr{L}_u$(Sp($u$)) and let $(x, y)\in D_f\times D_g$. The spectral measure of $(f(u)x, g(u)y)$ relative to $u$ is the measure $f g\cdot \mu_{x,y}$, where $\mu_{x,y}$ is the spectral measure of $(x, y)$ relative to $u$.

Let $\nu$ denote the spectral measure of $(f(u)x, g(u)y)$ relative to $u$. For every function $\varphi \in \mathscr{K}$ (Sp($u$)), one has

Sp($u$)

$=\langle f(u)x|(\varphi g)(u)y\rangle =\int f \varphi g \mu_{x,y}$,

Sp($u$)

(prop. 5, b)), whence $\nu =f g\cdot \mu_{x,y}$.