Let's say we have some integral, such that for a particular function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ $$\int_{\mathbb{R}^{n-m}} \int_{\mathbb{R}^m}f^+ - \int_{\mathbb{R}^{n-m}}\int_{\mathbb{R}^m}f^-$$ with $f^+:=max(f,0),f^-:=max(-f,0)$ and the following integrals are defined.
How do I see that this is the same as: $$\int_{\mathbb{R}^{n-m}} (\int_{\mathbb{R}^m}f^+-\int_{\mathbb{R}^m}f^-)^+ - \int_{\mathbb{R}^{n-m}}(\int_{\mathbb{R}^m}f^+-\int_{\mathbb{R}^m}f^-)^-$$
if I know that my integral is invariant under different decompositions. So my question is: Is there an easy argument, why both decompositions define the same function and both integrals consequently coincide? ( The outer plus and minus in the second integral are defined in the same way as they were earlier). If anything is unclear, please let me know.