The monotone convergence theorem shows the integral wants to be linear. This is the essential content of the next theorem. We can’t say it is linear yet because to be linear, something must be defined on a vector space or something similar where it makes sense to consider linear combinations and the integral has only been defined at this point on nonnegative measurable functions.
Proof: By Theorem 19.1.6 on Page 1422 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 20.2.3,
As long as you are allowing functions to take the value +∞, you cannot consider something like f +