If $f=f_{1}-f_{2}$ with $f_{1},f_{2}\geq0$ a.e. then $f_{1}=f^{+}$ and $f_{2}=f^{-}$? (measure theory)

Question

In measure theory the decomposition $f=f^{+}-f^{-}$ for a measurable function $f$ is commonly used, where $f^{+}=\max(0,f)\geq0$ and $f^{-}=\max(0,-f)\geq0$. Now suppose that $f=f_{1}-f_{2}$ with $f_{1}\geq0$ and $f_{2}\geq0$ almost everywhere. Can we conclude that $f_{1}=f^{+}$ and $f_{2}=f^{-}$? I don't think this need to be true, but I cannot come up with a counter example. Any suggestions are greatly appreciated