32.9. THE INTEGRAL’S RIGHTEOUS ALGEBRAIC DESIRES 647

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

Theorem 32.8.2 (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.

Proof: Let gn(ω) = inf{ fk(ω) : k ≥ n}. Then

g−1n ([a,∞]) = ∩∞

k=n f−1k ([a,∞]) ∈ F

Thus gn is measurable. Now the functions gn form an increasing sequence of nonnegativemeasurable functions. Thus g−1 ((a,∞)) = ∪∞

n=1g−1n ((a,∞)) ∈ F so g is measurable also.

By monotone convergence theorem,∫gdµ = lim

n→∞

∫gndµ ≤ lim inf

n→∞

∫fndµ.

The last inequality holding because∫gndµ ≤

∫fndµ.

(Note that it is not known whether limn→∞

∫fndµ exists.)

32.9 The Integral’s Righteous Algebraic DesiresThe monotone convergence theorem shows the integral wants to be linear. This is theessential content of the next theorem.

Theorem 32.9.1 Let f ,g be nonnegative measurable functions and let a,b be non-negative numbers. Then a f +bg is measurable and∫

(a f +bg)dµ = a∫

f dµ +b∫

gdµ. (32.13)

Proof: By Theorem 32.2.8 on Page 636 there exist increasing sequences of nonnegativesimple functions, sn → f and tn → g. Then a f +bg, being the pointwise limit of the simplefunctions asn+btn, is measurable. Now by the monotone convergence theorem and Lemma32.7.3, ∫

(a f +bg)dµ = limn→∞

∫asn +btndµ = lim

n→∞

(a∫

sndµ +b∫

tndµ

)= a

∫f dµ +b

∫gdµ.