The monotone convergence theorem shows the integral wants to be linear. This is the essential content of the next theorem.
Proof: By Theorem 7.1.6 on Page 445 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 8.2.3,
As long as you are allowing functions to take the value +∞, you cannot consider something like f +