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However, I think this space is too artifitial and it's better to work with signed Radon measures locally. So, Hahn decomposition theorem holds as you has pointed out
Hahn decomposition theorem states that each signed measure is $\mu=\mu^+-\mu^-$ with $\mu^+,\mu^-$ mutually singular positive measures, one of them is finite. But the space of signed measures of my first comment was: $$\{\mu-\nu:\mu,\nu\mbox{ are positive measures}\}$$ (with formal sums). I was not asuming the Hahn decomposition property on it and it can't be deduced from definition because elements of this space aren't real measures (only formal expresions designed for working locally on compacts). If I want this property I have to asume it in the definition.
I say "problems" because $\mu^+,\mu^-$ seem to be too arbitrary. But perhaps there are not better descriptions. I'll go on reading. Thank you very much!
I was afraid of changing the standard definition because I wanted to go on seeing this measures as distributions in the same way we see locally finite borel measures, but there is no problem. The only problem I have, if we use $\mathcal{M}^p$ for noting this subspace is to describe in an easy way the intersection $L^1_{loc}\cap \mathcal{M}^p$ (in the sense of distributions). Such functions $f$ verify: $$\int f\cdot\varphi \,dx=\int \phi \,d\mu^+ +\int \phi \,d\mu^-$$ for every $\varphi\in C^{\infty}_c$ and certain $\mu^+,\mu^-$ positive Radon measures
I agree with you. I'm reading about related topic in other references just know and it seems that they only consider meausures locally (defined for bounded Borel sets). So, what you mean is that the space of signed Radon measures they are problably considerying is kind of the vector space of formal sums $\alpha\cdot \mu+\beta\cdot \mu$ with $\mu,\nu$ positive Radon measures, isn't it?
