As I already pointed out in this answer to another recent question of yours, the dual of the Banach space $C_b(E)$ is not $\mathcal M(E)$ but the larger space $rba(E)$ of signed set functions that are regular, bounded, and additive, defined on the algebra $A_E$ generated by the open (or closed) sets of $E$. In other words, $rba(E)$ consists of all functions $\mu : \mathcal A_E \to \mathbb{R}$ satisfying the following conditions:

- $\mu(\varnothing) = 0$;
- $\mu$ is finitely additive (not necessarily $\sigma$-additive);
- $\mu$ is bounded;
- for all $A \in \mathcal A_E$ one has
\begin{align*}
\mu(A) &= \sup\{\mu(F) \, : \, F \subseteq A\ \text{closed}\} \\[1ex]
&= \inf\,\{\mu(V) \: : \: V \supseteq A \ \text{open}\}.
\end{align*}
(
**Note:** depending on the context, some authors use a different notion of regularity, where the closed sets $F$ in the supremum are replaced by compact sets. The results stated in this answer are no longer correct with this other notion of regularity.)

When equipped with the total variation norm, $rba(E)$ is isometrically isomorphic with $C_b(E)'$; see [DS58, Theorem IV.6.2 (p.262)] or [AB06, Theorem 14.10 (p.495)].
Your space $\mathcal M(E)$ embeds (isometrically) in $rba(E)$, since every finite signed Borel measure on a metric space is automatically regular; see [DS58, Exercise III.9.22 (p.170)] or [AB06, Theorem 12.5 (p.436)].
However, in general, $rba(E)$ is larger than $\mathcal M(E)$; see this answer.

The references mentioned here are the only functional analysis textbooks that I am aware of that prove this specific result, but maybe others know additional references. Note: although a proof written in the language of functional analysis might be more pleasant to read for some, I don't see how that would lead to a significant simplification. Basic results on dual pairs are not going to help you determine the dual of any particular Banach space; you still have to get your hands dirty. (The Riesz representation theorem doesn't prove itself, so to speak.)

For an overview of the norm duals of various Banach spaces related to measure theory, see [AB06, Table 14.1 (p.499)].

### References.

[DS58] Nelson Dunford, Jacob T. Schwartz, *Linear Operators, Part I: General Theory*, Interscience, 1958.

[AB06] Charalambos D. Aliprantis, Kim C. Border, *Infinite Dimensional Analysis, A Hitchhiker's Guide*, Third Edition, Springer, 2006.

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