Additivity of Lebesgue integral for non necessarily positive fucntions

Question

I know that $\int( f +g) d\mu =\int f d\mu+\int gd\mu $, for positive and measurable $ f,g$. How do I prove the same result for generic $f,g$? By de definition I know that $ \int fd\mu=\int f^+-\int f^-$, where $f^+,f^-$ are respectively the positive and negative part of $f$. I'm confused since: \begin{align} &\int (f+g) d\mu = \int(f+g)^+d\mu-\int(f+g)^-d\mu, \text{but} \\& (f+g)^+ \ne f^+ +g^+,(f+g)^- \ne f^- +g^- \end{align} So that I cannot use additivity for positive functions to solve this.

Answer 1

Let $h=f+g$. This equality can be also written as $h^+-h^-=f^+-f^-+g^+-g^-$. Hence we have $h^++f^-+g^-=f^++g^++h^-$. Since we know Lebesgue integral is additive for non negative functions:

$\int h^++\int f^-+\int g^-=\int f^++\int g^++\int h^-$

And hence:

$\int h=\int h^+-\int h^-=\int f^+-\int f^-+\int g^+-\int g^-=\int f+\int g$