9.3.8 The Righteous Algebraic Desires Of The Lebesgue Integral
The monotone convergence theorem shows the integral wants to be linear. This is the
essential content of the next theorem.
Theorem 9.3.19 Let f,g be nonnegative measurable functions and let a,b be
nonnegative numbers. Then
Proof: By Theorem 9.3.9 on Page 644 there exist sequences of nonnegative simple
functions, sn → f and tn → g. Then by the monotone convergence theorem and Lemma
As long as you are allowing functions to take the value +∞, you cannot consider
something like f +
and so you can’t very well expect a satisfactory statement about
the integral being linear until you restrict yourself to functions which have values in a
vector space. This is discussed next.