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.
