Let $\Omega$ be a locally compact and Hausdorff topological space. The Riesz representation theorem says that $C_0(\Omega)^*$ , dual of the commutative C*-algebra $C_0(\Omega)$, is just the space of complex Radon measures $M(\Omega)$.

$$\gamma: M(\Omega)\simeq C_0(\Omega)^* : \gamma(\mu)(f)=\int f d\mu$$

Let $\mu$ be a complex Radon measure and denote $[\mu]$, by the total variation of $\mu$.

Question: It seems that $\gamma([\mu])=|\gamma(\mu)|$, where $|\gamma(\mu|)$ is the absolute value of the bounded linear functional $\gamma(\mu)$ on $C_0(\Omega)$ ?

1more comment