Fix a function $f\in L^1_\text{loc}(\mathbb{R^n})$.  Let
$$
L^1_\text{rel}[f]=\{ g\in L^1_\text{loc}(\mathbb{R}^n) : \|g-f\|_1<\infty\}.$$
be space of functions which differ from $f$ by an $L^1$ function. Observe, that for any $g\in L^1_\text{rel}[f]$ there is a well defined notion of relative integral given by
$$
I_\text{rel}(g; f)= \lim_{R\to \infty} \left(\int_{B_R} g\; dL^n- \int_{B_R} f\; dL^n\right).
$$
To see this observe, that for any $\varepsilon>0$, there an $R_\varepsilon$ so that
$$
\int_{\mathbb{R}^n\setminus B_{R_\varepsilon}} |f-g| \; dL^n <\varepsilon
$$
and so for $R_\varepsilon<R_1<R_2$, 
$$
 \left|\int_{B_{R_1}} g\; dL^n- \int_{B_{R_1}} f\; dL^n- \left(\int_{B_{R_2}} g\; dL^n- \int_{B_{R_2}} f\; dL^n \right)\right|=\left|\int_{B_{R_2}\setminus B_{R_1}}f-g\; dL^n \right| < \varepsilon.
$$

My question is what extent is there something analogous when one considers  Radon measures on $\mathbb{R}^n$.  For instance, the above means this should be possible for measures which are absolutely continuous with respect to Lebesgue measure (and more generally absolutely continuous outside of a compact set).  To what extent can one weaken this?

Any references would be appreciated.