Remark

Let A be a commutative unital stellar subalgebra of $\mathscr{L}(E)$. For every $x$ and $y$ in E, let $\mu_{x,y}$ denote the spectral measure of $(x, y)$ relative to A. The mapping defined by $(x, y)\mapsto \mu_{x,y}$ of $E\times E$ into $\mathscr{M}^1(\mathsf{X}(A))$ is sesquilinear.

Let $u$ be a normal endomorphism of E and A the unital stellar subalgebra of $\mathscr{L}(E)$ generated by $u$. It is commutative. The space $\mathsf{X}(A)$ is identified with Sp($u$) by the mapping $\chi \mapsto \chi (u)$ (lemma 10 of I, p. 109). The spectral measure $\mu_{x,y}$ of $(x, y)$ relative to A is therefore identified with a measure on Sp($u$), called the spectral measure of $(x, y)$ relative to $u$. For every function $f\in \mathscr{C}$ (Sp($u$)), one then has

Sp($u$)

by def. 5 of I, p. 110.