I am looking at the following version of the Riesz representation theorem:
Let $X$ be a compact metric space and let $\Lambda : C(X) \to \mathbb R$ (or $\mathbb C$) be a continuous linear functional. Then there exists a unique positive measure $|\mu|$ and a measurable function $g$ with $\|g\|_\infty =1$ such that for all $f \in C(X)$: $$ \Lambda (f) = \int_X f g d |\mu|$$
My question is, could I instead say that there exists a unique complex signed mesure $\mu$ such that $$ \Lambda (f) = \int_X f d \mu$$
If yes, why do we want to write signed measures as a product of a bounded measurable function with a positive measure? If no, why not?
Thanks for your help.