I am slightly confused about two different approaches to arrive at a conjugate of a complex measure. Any hints and/or clarifications on how these notions are related would be great. (It is also possible that I have some errors in the following lines of reasoning):
First attempt:
If $\mu$ is a complex measure, then define the complex measure $\overline{\mu}$ via $$\overline{\mu}(B):= \overline{\mu(B)}$$ Now if this attempt is pursued, I am interested in the precise relation when integrating a bounded measurable function $h$, i.e. is
$$\int f \mathrm{d}\overline{\mu}= \int \overline{f} \mathrm{d}\mu$$ true? If so, then why?
Further, if it is true, how does this relate to the following implication obtained via Radon-Nykodym: Both measures are dominated by the total variation measure $|\mu|$, i.e. there are functions $h_1, h_2$ such that: $$\int f \mathrm{d}\overline{\mu} = \int f h_1 \mathrm{d}|\mu|$$ and $$\int f \mathrm{d}\mu = \int f h_2 \mathrm{d}|\mu|$$ hold. What is the precise relation between $h_1$ and $h_2$?
Second attempt:
View the original complex measure $\mu$ as a continuous linear functional on $C_0(X)$ ( X denoting a locally compact Hausdorff-space). Hence, for any member $f \in C_0(X)$ integration via $\mu$ defines a functional $\Phi(f)$. Now, if we apply the operation of complex conjugation to the output, i.e. we define $\Phi_2(f):= \overline{\Phi(f)}$ we should still have a continuous linear functional. Then there is a complex Borel-measure $\eta$ which represents this functional. Is the labeling $\overline{\mu}:= \eta$ consistent with the defintion given in attempt 1? If so, why?