To put this result into context, I will show how to deduce it from the big theorems. (This is not nearly as elementary as Kavi's proof.)
Unfortunately, different authors use different notation / terminology, so let me set up a few definitions first.
Definition. A finitely additive signed measure (or charge) is a function $\mu : \mathcal A \to [-\infty,\infty]$ with the following properties:
- $\mu(\varnothing) = 0$;
- $\mu$ assumes at most one of the values $-\infty$ and $+\infty$;
- $\mu$ is finitely additive.
Comparted to signed measures, the requirement of $\sigma$-additivity is weakened to finite additivity.
If $\Omega$ is a topological space, let $\mathcal A_\Omega$ and $\mathcal B_\Omega$ denote respectively the algebra and the $\sigma$-algebra generated by the open (or closed) sets of $\Omega$. Then $\mathcal A_\Omega \subseteq \mathcal B_\Omega$, and $\mathcal B_\Omega$ is the $\sigma$-algebra generated by $\mathcal A_\Omega$, so every measure on $B_\Omega$ is uniquely determined by its values on $A_\Omega$.
Definition. Let $\Omega$ be a topological space. We say that a finitely additive signed measure $\mu$ on $\mathcal A_\Omega$ is regular if for every $A \in \mathcal A_\Omega$ 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. Different authors use different notions of regularity. In particular, sometimes the closed sets $F \subseteq A$ are replaced by compact sets.
We will use the following well-known results:
Theorem. Let $\Omega$ be a normal Hausdorff space. Then $C_b(\Omega)'$ is isometrically isomorphic to the space $rba(\Omega)$ of all regular, finitely additive signed measures of bounded variation on $\mathcal A_\Omega$, equipped with the total variation norm.
See [DS58, Theorem IV.6.2 (p.262)] or [AB06, Theorem 14.10 (p.495)].
For general topological spaces, see this question on MathOverflow.
Lemma. Every finite signed Borel measure on a metric space is regular.
See [DS58, Exercise III.9.22 (p.170)] or [AB06, Theorem 12.5 (p.436)], among others.
(Side note: this is not true for the other notion of regularity, with compact sets instead of closed sets, as can be seen from this answer on MathOverflow.)
It follows that the space $\mathcal M(\Omega)$ of finite signed Borel measures is a subspace of $rba(\Omega) \cong C_b(\Omega)'$. To complete the proof, note that every normed space $X$ separates points on every subspace of $X'$: if $\varphi(x) = 0$ for all $x \in X$, then $\varphi = 0$.
(More generally, for a bilinear pairing $\langle E , F \rangle$ to be non-degenerate, so that it is a proper dual pairing, it is necessary and sufficient that the induced maps $E \to F^*$ and $F \to E^*$ are injective.)
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.
Best Answer
I suppose that both signed measures are finite, since otherwise we cannot expect that the integral over arbitrary continuous functions exists. You can approximate any interval $[a,b]$ by (for example) the sequence $$g_n(x) := \begin{cases} 1 & \text{ if } x \in [a,b] \\ nx + (1-na) &\text{ if } x \in [a-1/n,a] \\ -nx +(1+nb) & \text{ if } x \in [b,b+1/n] \\ 0 & \text{otherwise} \end{cases}.$$ By the dominated convergence theorem (for signed measures) we get $$\nu_1([a,b]) =\nu_2([a,b]).$$ Thus, both measures are equal on a $\cap$-stable generator of the Borel-$\sigma$-algebra. Now, you need to extend the uniqueness theorem for $\sigma$-finite measures. In fact, the same proof applies. Note that $$\mathcal{C}=\{B\in\mathcal{B}_{[-\pi, \pi]} : \nu_1(B)=\nu_2(B)\}$$ is a $\pi$-system containing a $\cap$-stable generator with a sequence $E_n \in \mathcal{C}$ such that $E_n \uparrow \Omega$.
For completeness, we should give reasons for the application of the dominated convergence theorem: By Jordan's decomposition theorem, you can write for a signed measure $\mu = \mu_1 - \mu_2$ with measures $\mu_1$ and $\mu_2$. (Moreover, there exists a 'minimal decomposition' - called the Jordan decompisition.)