14.9. THE MONOTONE CONVERGENCE THEOREM 337

where the sets Ai∩B j are disjoint and measurable. By Lemma 14.8.2,∫as+btdµ =

m

∑j=1

n

∑i=1

(aα i +bβ j)µ(Ai∩B j)

=n

∑i=1

am

∑j=1

α iµ(Ai∩B j)+bm

∑j=1

n

∑i=1

β jµ(Ai∩B j)

= an

∑i=1

α iµ(Ai)+bm

∑j=1

β jµ(B j) = a∫

sdµ +b∫

tdµ .

14.9 The Monotone Convergence TheoremThe following is called the monotone convergence theorem also Beppo Levi’s theorem.This theorem and related convergence theorems are the reason for using the Lebesgueintegral. If limn→∞ fn (ω) = f (ω) and fn (ω) is increasing in n, then clearly f is alsomeasurable because

f−1 ((a,∞]) = ∪∞k=1 f−1

k ((a,∞]) ∈F

The version of this theorem given here will be much simpler than what was done with thegeneralized Riemann integral and it will be easier to state and remember and use.

Theorem 14.9.1 (Monotone Convergence theorem) Let f have values in [0,∞] andsuppose { fn} is a sequence of nonnegative measurable functions having values in [0,∞]and satisfying

limn→∞

fn(ω) = f (ω) for each ω.

· · · fn(ω)≤ fn+1(ω) · · ·Then f is measurable and ∫

f dµ = limn→∞

∫fndµ.

Proof: By Lemma 14.7.2 and interchange of supremums,

limn→∞

∫fndµ = sup

n

∫fndµ

= supn

suph>0

∞

∑k=1

µ ([ fn > kh])h = suph>0

supN

supn

N

∑k=1

µ ([ fn > kh])h

= suph>0

supN

N

∑k=1

µ ([ f > kh])h = suph>0

∞

∑k=1

µ ([ f > kh])h =∫

f dµ.

The next theorem, known as Fatou’s lemma is another important theorem which justi-fies the use of the Lebesgue integral.

Theorem 14.9.2 (Fatou’s lemma) Let fn be a nonnegative measurable function. Letg(ω) = liminfn→∞ fn(ω). Then g is measurable and∫

gdµ ≤ lim infn→∞

∫fndµ .

In other words, ∫ (lim inf

n→∞fn

)dµ ≤ lim inf

n→∞

∫fndµ