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

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.