Hello,


As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu\_+$ and negative one $\mu\_-$ by [the Hahn decomposition theorem.](http://en.wikipedia.org/wiki/Hahn_decomposition_theorem)


My question is whether each real-valued Radon measure $\nu$ on $R$ can be decomposed by the positive and negative parts? I think it is true. If the proof is long, Where can I find a reference?


Here is my motivation: the real-valued Radon measure on $R$ seems more general than the signed measure. For example, on the [wikipedia](http://en.wikipedia.org/wiki/Radon_measure), you can find the example $sin(x)d x$ which is a real-valued Radon measure but not a signed measure. By this reason, I would like to work under the real-valued radon measure. However, I am not an expert on measure theory, especially on Radon measure stuff. 


Thank you very much!


Anand



Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures?
I'd take the view that a real-valued Radon measure "really is" a continuous linear functional $\Lambda:C\_{00}(X,\mathbb R) \rightarrow \mathbb R$. Here $X$ is a locally compact space, and $C\_{00}(X,\mathbb R)$ is the collection of compactly supported continuous real valued functions on $X$. By "continuous", we mean that for each compact $K\subseteq X$, there is a constant $c>0$ such that
$$ |\Lambda(f)| \leq c \|f\|\_\infty $$
for all continuous $f$ supported in $K$. (For example, with $\Lambda$ given by integrating against $\sin(x) \ dx$, we can take $c$ to be the Lebesgue measure of $K$).


So you could just work abstractly with such linear functionals. (This is going to be a sketch, but if you've done a course on abstract integration, nothing should be too great a surprise). Then you do the usual trick: define
$$ \Lambda\_+(f) = \sup\{ \Lambda(g) : 0\leq g\leq f \} $$
for all $f\in C\_{00}(X,\mathbb R)$ which are (pointwise) positive. The supremum is positive (take $g=0$) and finite because $\Lambda$ is continuous.
Notice that $\Lambda\_+$ is positive homogeneous, and clearly $\Lambda\_+(f\_1) + \Lambda\_+(f\_2) \leq \Lambda\_+(f\_1+f\_2)$. Conversely, if $0 \leq g \leq f\_1+f\_2$, then let $g\_1=\min(g,f\_1)$ and $g\_2=g-g\_1$, so $0\leq g\_1\leq f\_1$, $0\leq g\_2\leq g\_2$, and $\Lambda(g) = \Lambda(g\_1)+\Lambda(g\_2)$. It follows that $\Lambda\_+$ is additive.


Then extend $\Lambda\_+$ to all of $C\_{00}(X,\mathbb R)$ by $\Lambda\_+(f) = \Lambda\_+(f\_+) - \Lambda\_+(f\_-)$. That $\Lambda\_+$ is linear will follow from *exactly* the same proof which shows that the Lebesgue integral is linear (having first defined it for positive functions) from pretty much any course on integration.


Then define $\Lambda\_-(f) = \Lambda\_+(f) - \Lambda(f)$, so
$$ \Lambda\_-(f) = \sup\{ \Lambda(h) : -f\leq h\leq 0 \}\geq 0 $$
(set $h=g-f$). So $\Lambda\_+$ and $\Lambda\_-$ are *positive* linear functionals on $C\_{00}(X,\mathbb R)$ and are hence given by positive Radon measures.


I don't know a self-contained reference-- as I say, I was exposed to these ideas in courses on integration theory, and basic functional analysis. I like Rudin's book "Real and complex analysis" for the basics on positive Radon measures.


