Kenneth Kuttler

210 CHAPTER 9. THE LEBESGUE INTEGRAL

It is clear from the construction that |s| ≤ | f |.What about 9.4? Let θ ∈ C be such that |θ | = 1 and θ

∫f dµ = |

∫f dµ| . Then from

what was shown above about the integral being linear,∣∣∣∣∫ f dµ

∣∣∣∣= θ

∫f dµ =

∫θ f dµ =

∫Re(θ f )dµ ≤

∫| f |dµ.

If f ,g ∈ L1 (Ω) , then it is known that for a,b scalars, it follows that a f +bg is measur-able. See Lemma 9.7.2. Also

∫|a f +bg|dµ ≤

∫|a| | f |+ |b| |g|dµ < ∞. ■

The following corollary follows from this. The conditions of this corollary are some-times taken as a definition of what it means for a function f to be in L1 (Ω).

Corollary 9.7.7 f ∈ L1(Ω) if and only if there exists a sequence of complex simplefunctions, {sn} such that

sn (ω)→ f (ω) for all ω ∈Ω

limm,n→∞

∫(|sn− sm|)dµ = 0 (9.7)

When f ∈ L1 (Ω) , ∫f dµ ≡ lim

n→∞

∫sn. (9.8)

Proof: From the above theorem, if f ∈ L1 there exists a sequence of simple functions{sn} such that ∫

| f − sn|dµ < 1/n, sn (ω)→ f (ω) for all ω

Then∫|sn− sm|dµ ≤

∫|sn− f |dµ +

∫| f − sm|dµ ≤ 1

n +1m .

Next suppose the existence of the approximating sequence of simple functions. Thenf is measurable because its real and imaginary parts are the limit of measurable functions.By Fatou’s lemma,

∫| f |dµ ≤ liminfn→∞

∫|sn|dµ < ∞ because |

∫|sn|dµ−

∫|sm|dµ| ≤∫

|sn− sm|dµ which is given to converge to 0. Thus {∫|sn|dµ} is a Cauchy sequence and

is therefore, bounded.In case f ∈ L1 (Ω) , letting {sn} be the approximating sequence, Fatou’s lemma implies∣∣∣∣∫ f dµ−

∫sndµ

∣∣∣∣≤ ∫ | f − sn|dµ ≤ lim infm→∞

∫|sm− sn|dµ < ε

provided n is large enough. Hence 9.8 follows. ■This is a good time to observe the following fundamental observation which follows

from a repeat of the above arguments.

Theorem 9.7.8 Suppose Λ( f ) ∈ [0,∞] for all nonnegative measurable functionsand suppose that for a,b≥ 0 and f ,g nonnegative measurable functions,

Λ(a f +bg) = aΛ( f )+bΛ(g) .

In other words, Λ wants to be linear. Then Λ has a unique linear extension to the set ofmeasurable functions { f measurable : Λ(| f |)< ∞} , this set being a vector space.

210 CHAPTER 9. THE LEBESGUE INTEGRALIt is clear from the construction that |s| < |f].What about 9.4? Let 6 € C be such that |6| = 1 and 0 f fdu =|f fdu|. Then fromwhat was shown above about the integral being linear,| rau] =6 [sau foran= [Re(ofrau< [planIf f,¢ € L' (Q), then it is known that for a, scalars, it follows that af + bg is measur-able. See Lemma 9.7.2. Also [laf +bg|du < fa||f|+ |b| |g|\du <0. iThe following corollary follows from this. The conditions of this corollary are some-times taken as a definition of what it means for a function f to be in L! (Q).Corollary 9.7.7 f € L'(Q) if and only if there exists a sequence of complex simplefunctions, {s,} such thatSn(@) > f (@) for alla EQlim noo f (|Sn —Sm|)du —0 (9.7)When f € L'(Q),/ fa = tim [ sy. (9.8)Proof: From the above theorem, if f € L! there exists a sequence of simple functions{s,} such that[ipa oldu <1/n, s,(@) + f(@) for all @Then ||sn—snldi < flon—fldu+S|f—smldu <t+e.Next suppose the existence of the approximating sequence of simple functions. Thenf is measurable because its real and imaginary parts are the limit of measurable functions.By Fatou’s lemma, f{|f|du < liminf,,.. f |s,|du < ce because | |s,|du — f |sn|du| <J |sn —Sm|dp which is given to converge to 0. Thus {f |s,|du} is a Cauchy sequence andis therefore, bounded.Incase f € L! (Q), letting {s,} be the approximating sequence, Fatou’s lemma implies[ran [sna < [lf slau <tim inf [lsm—snldp <eprovided n is large enough. Hence 9.8 follows.This is a good time to observe the following fundamental observation which followsfrom a repeat of the above arguments.Theorem 9.7.8 suppose A(f) € [0,c> for all nonnegative measurable functionsand suppose that for a,b > 0 and f ,g nonnegative measurable functions,A(af +bg) =aA(f)+bA(g).In other words, A wants to be linear. Then A has a unique linear extension to the set ofmeasurable functions {f measurable : A(|f|) < °°}, this set being a vector space.