Let $f:\mathbb{R}\to\mathbb{R}$ be a right-continuous function that is locally of bounded variation (i.e., of bounded variation on every $[a,b]\subset\mathbb{R}$; henceforth, "LBV function"), then there exists a unique (signed) measure $\mu$ on $\mathscr{B}(\mathbb{R})$ (the Borel subsets of $\mathbb{R}$) such that $\mu((a,b]) = f(b) - f(a)$ for every $a<b$. I'll denote this measure as $R_f$. Likewise, if $f$ is a LBV left-continuous function, then there exists a unique (signed) measure $L_f$ on $\mathscr{B}(\mathbb{R})$ such that $L_f([a,b)) = f(b) - f(a)$ for every $a<b$.
Now, suppose that $f:\mathbb{R}\to\mathbb{R}$ is the sum of a LBV right-continuous function $g$ and a LBV left-continuous function $h$, then it is very natural to "assign" a measure to $f$, namely $R_g + L_h$. This assignment is well-defined: If $\tilde{g}$ is a LBV right-continuous function and $\tilde{h}$ is a LBV left-continuous function such that $f = g+h = \tilde{g}+\tilde{h}$, then $g-\tilde{g} = \tilde{h}-h$ is a LBV continuous function, so $R_\tilde{g} + L_\tilde{h} = R_g - R_{g-\tilde{g}} + L_h + L_{\tilde{h}-h} = R_g + L_h$. If we think LBV right-continuous functions and LBV left-continuous functions as "nice", then their sums should be nice too.
So my question is: is there an easy criterion to determine whether a function is the sum of a LBV right-continuous function and a LBV left-continuous function, and how should we find a decomposition?