Scalar-measure construction of a projection-valued measure

For a unital star-representation $\pi :C(K)\to \mathcal{B}(H)$, the Riesz-Markov-Kakutani representation theorem gives regular measures ${\mu }_{x,y}$ representing $f\mapsto ⟨\pi (f)x,y⟩$. Define $P(E)$ by $⟨P(E)x,y⟩={\mu }_{x,y}(E)$. Positivity and ${\mu }_{x,x}(K)=\Vert x{\Vert }^2$ make these positive contractions. The identities ${\mu }_{\pi (f)x,y}=f{\mu }_{x,y}$ imply commutation with $\pi (f)$ and ${\mu }_{P(E)x,y}=1_E{\mu }_{x,y}$; hence $P(F)P(E)=P(F\cap E)$. Thus each $P(E)$ is an orthogonal projection. Scalar countable additivity and orthogonality give countable additivity in the strong operator topology, constructing the normalized regular projection-valued measure.