Proof. (1): Let $x, y \in H$, $\seqf{B_j}\subset \cb_{X}$ be disjoint Borel sets, and $B = \bigsqcup_{j = 1}^{n} B_{j}$, then for each $1 \le i < j \le n$, $E(B_{i})(H) \perp E(B_{j})(H)$, so by the Cauchy-Schwarz inequality and the Pythagorean Theorem,

As the above holds for all finite sequences of disjoint Borel sets, $\norm{E_{x, y}}_{\text{var}}\le \norm{x}_{H} \norm{y}_{H}$.

(2): For each $x, y \in H$ and $B, C \in \cb_{X}$,

By linearity, $fdE_{x, y}\in \mathscr{E}$ for all $f \in \Sigma(X; \complex)$. For each $f \in \Sigma(X; \complex)$, the mapping $\mu \mapsto f d\mu$ is continuous in the total variation norm, so $fd\mu \in \mathscr{E}$ for all $\mu \in \mathscr{E}$ and $f \in \Sigma(X; \complex)$. By Proposition 16.1.10, $\Sigma(X; \complex)$ is dense in $L^{1}(\mu; \complex)$ for all $\mu \in \mathscr{E}$. Therefore $fd\mu \in \mathscr{E}$ for all $f \in L^{1}(\mu; \complex)$ and $\mu \in \mathscr{E}$.

Finally, let $\mu \in \mathscr{E}$ and $\nu \in M_{R}(X; \complex)$ with $\nu \ll \mu$, then by the Radon-Nikodym Theorem, there exists $f \in L^{1}(\mu; \complex)$ such that $d\nu = f d\mu \in \mathscr{E}$.

(3): By (2), for any $\mu \in \mathscr{E}$ and $f \in L^{1}(\mu; \complex)$, $fd\mu \in \mathscr{E}$ as well. By Proposition 38.8.1, there exists a unique weak*-continuous involution and separately weak*-continuous product on $\mathscr{E}^{*}$ making $\mathscr{E}^{*}$ a commutative unital $C^{*}$-algebra, and $J|_{C(X; \complex)}$ a unital *-homomorphism. Since

1. (i)

   $J$ is $\sigma(B^{\infty}(X; \complex), M_{R}(X; \complex))$-$\sigma(\mathscr{E}^{*}, \mathscr{E})$ continuous.

2. (ii)

   Conjugation on $B^{\infty}(X; \complex)$ is $\sigma(B^{\infty}(X; \complex), M_{R}(X; \complex))$-continuous.

3. (iii)

   Multiplication on $B^{\infty}(X; \complex)$ is separately $\sigma(B^{\infty}(X; \complex), M_{R}(X; \complex))$-continuous.

the mapping $J$ is a unital *-homomorphism.$\square$