Kenneth Kuttler

284 CHAPTER 10. THE ABSTRACT LEBESGUE INTEGRAL

increasing sequence of nonnegative measurable functions so the monotone convergencetheorem applies. This yields∫

gdµ = limn→∞

∫gndµ ≤ lim inf

n→∞

∫fndµ.

The last inequality holding because∫

gndµ ≤∫

fndµ. (Note that it is not known whetherlimn→∞

∫fndµ exists.) ■

10.6 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 10.6.1 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. (10.3)

Proof: By Theorem 9.1.6 on Page 239 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 Lemma10.2.3, ∫

(a f +bg)dµ = limn→∞

∫asn +btndµ = lim

n→∞

(a∫

sndµ +b∫

tndµ

)= a

∫f dµ +b

∫gdµ. ■

As long as you are allowing functions to take the value +∞, you cannot consider some-thing like f +(−g) and so you can’t very well expect a satisfactory statement about theintegral being linear until you restrict yourself to functions which have values in a vectorspace. To be linear, a function must be defined on a vector space. This is discussed next.

10.7 The Lebesgue Integral, L1

The functions considered here have values in C, which is a vector space. A function f withvalues in C is of the form f = Re f + i Im f where Re f and Im f are real valued functions.In fact Re f = f+ f

2 , Im f = f− f2i .

Definition 10.7.1 Let (Ω,S ,µ) be a measure space and suppose f : Ω→C. Thenf is said to be measurable if both Re f and Im f are measurable real valued functions.

Of course there is another definition of measurability which says that inverse images ofopen sets are measurable. This is equivalent to this new definition.

Lemma 10.7.2 Let f : Ω→ C. Then f is measurable if and only if Re f , Im f are bothreal valued measurable functions. Also if f ,g are complex measurable functions and a,bare complex scalars, then a f +bg is also measurable.

284 CHAPTER 10. THE ABSTRACT LEBESGUE INTEGRALincreasing sequence of nonnegative measurable functions so the monotone convergencetheorem applies. This yieldsJ sdu=jim f gnaw <tim int, | fra.The last inequality holding because f gndu < f f,du. (Note that it is not known whetherlimy yoo f fnd exists.) Hl10.6 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 10.6.1 Ler J .g be nonnegative measurable functions and let a,b be non-negative numbers. Then af + bg is measurable and[laf+be)du =a [fan +b | gap. (10.3)Proof: By Theorem 9.1.6 on Page 239 there exist increasing sequences of nonnegativesimple functions, s, — f and t, — g. Then af +bg, being the pointwise limit of the simplefunctions as, + bt,, is measurable. Now by the monotone convergence theorem and Lemma10.2.3,neo/ (af +bg)de = tim fas, + btyd = tim (« / snd +b / indi)nooa [ fay +b | gap. :As long as you are allowing functions to take the value +0, you cannot consider some-thing like f+(—g) and so you can’t very well expect a satisfactory statement about theintegral being linear until you restrict yourself to functions which have values in a vectorspace. To be linear, a function must be defined on a vector space. This is discussed next.10.7 The Lebesgue Integral, L!The functions considered here have values in C, which is a vector space. A function f withvalues in C is of the form f = Re f + iIm f where Re f and Im f are real valued functions.In fact Re f = &E, Im f = 54.Definition 10.7.1 Lez (Q,.%, UW) be a measure space and suppose f :Q— C. Thenf is said to be measurable if both Re f and Im f are measurable real valued functions.Of course there is another definition of measurability which says that inverse images ofopen sets are measurable. This is equivalent to this new definition.Lemma 10.7.2 Let f:Q—C. Then f is measurable if and only if Re f Im f are bothreal valued measurable functions. Also if f,g are complex measurable functions and a,bare complex scalars, then af + bg is also measurable.