The following is a part of a theorem of Folland's book:
Let $X$ be a compact space, $B(X)$ the space of bounded Borel measurable functions on $X$, and $C(X)$ the space of continuous function on $X$.
Suppose for $f\in C(X)$, $T_f$ is the inverse Gelfand transform of $f$ in the commutative C*-subalgebra $A$ of $B(H)$.
For $\xi,\eta\in H$, define the map $\phi_{\xi,\eta}: C(X)\to \Bbb C$ such that $\phi_(\xi,\eta)(f) = \langle T_f\xi,\eta\rangle$. Also there is a unique Boreal measure $\mu_{\xi,\eta}\in M(X)$ such that $\langle T_f \xi , \eta \rangle = \int f d\mu_{\xi,\eta} $.
The map $f\to T_f$ gives a representation of the algebra $C(X)$ as bounded operators on $H$. We now extend this representation to the larger algebra $B(X)$. if $f\in B(X)$, then $$|\int fd\mu_{\xi,\eta}|\leq ||f||||\xi||||\eta||$$ Hence there is a unique $T_f\in B(H)$ such that $\langle T_f\xi,\eta\rangle =\int f d\mu_{\xi,\eta}$.
But I do not know how he concludes the last line. Please help me. Thanks.