Proof of the integration of L^1 functions

I am having difficulty proving the following,

Let

Prove that

Background: is the set of complex measurable functions f such that

I will also be using and and a lemma that if f is a positive measurable function then

My Proof thus far: (I have separated the proof into 2 cases)

Case 1:

and this is where I am stuck both so I can apply the lemma stated earlier but I can not split the integral unless both functions are positive but is negative.

Any suggestions on how i can get around this obstacle would be greatly appreciated !