The monotone convergence theorem shows the integral wants to be linear. This is the essential content of the next theorem.
Proof: By Theorem 5.1.9 on Page 412 there exist increasing sequences of nonnegative simple functions, sn → f and tn → g. Then af + bg, being the pointwise limit of the simple functions asn + btn, is measurable. Now by the monotone convergence theorem and Lemma 6.2.3,
As long as you are allowing functions to take the value +∞, you cannot consider something like f +