Partitioning $mathbb{R}$ for given functions.

Let $f,g$ be two real valued bounded measurable functions. I’d like to construct two Borel sets $O_1$ and $O_2$ of $mathbb{R}$ such that they partition $mathbb{R}$ ($O_1 cup O_2= mathbb{R}$ and $O_1 cap O_2= phi$) and satisfy $ (supp(f-g)^{+})^c subset O_1$ and $ (supp(f-g)^{-})^c subset O_2$. Where $f^pm$ denotes the positive and negative part of $f$ respectively and supp stands for the support of the function.

If $f,g$ were continuous then we could have chosen $O_1=supp(f-g)^{-}$ and $O_2=supp(f-g)^{+}$. As in this case both $O_1$ and $O_2$ will be Borel sets. How do we choose $O_1,O_2$ for any bounded function?