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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.

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    I wasn't sure whether to add the soft-question tag or not, the question seems sort of soft.2012-08-19
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    No, it's probably nonsense. Sorry. :-)2012-08-19
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    @Thomas I'm glad : )2012-08-19
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    Yes, you could say that. However, to get from your second version to the first version, there's some work to do: there's a variant of Radon-Nikodym hidden in the statement, and your first version also gives the [Hahn decomposition theorem](http://en.wikipedia.org/wiki/Hahn_decomposition_theorem) essentially for free.2012-08-19

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