Kenneth Kuttler

7.7. THE LEBESGUE INTEGRAL, L1 165

Proof: By Theorem 6.1.10 on Page 140 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 Lemma7.2.3, ∫

(a f +bg)dµ = limn→∞

∫asn +btndµ

= limn→∞

(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 expect a satisfactory statement about the integral beinglinear until you restrict yourself to functions which have values in a vector space. To belinear, a function must be defined on a vector space. This is discussed next.

7.7 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

We first define the integral of real valued functions and then the integral of a complexvalued function will be of the form∫

f dµ =∫

Re( f )dµ + i∫

Im( f )dµ

Definition 7.7.1 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.

As is always the case for complex numbers, |z|2 = (Rez)2 +(Imz)2. Also, for g a realvalued function, one can consider its positive and negative parts defined respectively as

g+ (x)≡ g(x)+ |g(x)|2

, g− (x) =|g(x)|−g(x)

Thus |g| = g+ + g− and g = g+ − g− and both g+ and g− are measurable nonnegativefunctions if g is measurable. This follows because of Theorem 6.1.6. The mappings x→x+,x→ x− are clearly continuous. Thus g+ is the composition of a continuous functionwith a measurable function.

Then the following is the definition of what it means for a complex valued function fto be in L1 (Ω).

Definition 7.7.2 Let (Ω,F ,µ) be a measure space. Then a complex valued mea-surable function f is in L1 (Ω) if ∫

| f |dµ < ∞.

7.7. THE LEBESGUE INTEGRAL, L! 165Proof: By Theorem 6.1.10 on Page 140 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 Lemma7.2.3,[laf +bs)du = lim | as,+bt,dunoo ,= im (4 [sna +0 [ nds)a | fau+b f edu. |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 expect a satisfactory statement about the integral beinglinear until you restrict yourself to functions which have values in a vector space. To belinear, a function must be defined on a vector space. This is discussed next.7.7 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+ilm/f where Re f and Im f are real valued functions.In factf+f ffRe f = ——, Imf = .i er ayWe first define the integral of real valued functions and then the integral of a complexvalued function will be of the form[fan = [Re(pdu+i [im(fyduDefinition 7.7.1 Lez (Q,.%, UL) 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.As is always the case for complex numbers, |z|* = (Rez)* + (Imz)*. Also, for g a realvalued function, one can consider its positive and negative parts defined respectively asBix) + 1g - §\X)|— 8+( (x) BO Gy = Bo! (x)x)= 5 5Thus |g] = g* +g and g=g' —g~ and both g* and g™ are measurable nonnegativefunctions if g is measurable. This follows because of Theorem 6.1.6. The mappings x >xt,x + x7 are clearly continuous. Thus g* is the composition of a continuous functionwith a measurable function.Then the following is the definition of what it means for a complex valued function fto be in L! (Q).Definition 7.7.2 Ler (Q,F,U) be a measure space. Then a complex valued mea-surable function f is in L' (Q) if[flan <=.