The proposition and the first part of its proof are given below:
The book said "The theorem follows by establishing it separately for the positive and negative parts of $f$" but the book never did the proof for the negative part, could anyone help me establishing this proof?
If you prove the result for non-negative integrable functions $f$ you can apply it to $f^{+}$ as well as $f^{-}$ because these are non-negative integrable functions. Once you do this you can use the fact that $\int_E |f| =\int_E f^{+}+\int_E f^{-}$.