Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)\|$. Let $C_b(\Omega)=C_b(\Omega,\mathbb R)$. For a normal topological space $\Omega$ ( $T_4$-space) it holds that

$$ C_b(\Omega)^*=rba(\Omega), $$

where $rba(\Omega)$ is the space of regular bounded finitely additive measures, and also $$ x^*f=\int\limits_{\Omega}f(\omega)\mu(d\omega),\quad f\in C_b(\Omega),\quad x^*\in C_b(\Omega)^*,\quad \mu\in rba(\Omega) $$

Are there more precise results for the case $C_b (\mathbb R)$? Particularly I'm interested in more "beautiful" presentation of the measure $\mu$.