9.7. THE LEBESGUE INTEGRAL, L1 209

and the triangle inequality holds, ∣∣∣∣∫ f dµ

∣∣∣∣≤ ∫ | f |dµ. (9.4)

Also, for every f ∈ L1 (Ω) it follows that for every ε > 0 there exists a simple function ssuch that |s| ≤ | f | and

∫| f − s|dµ < ε.

Proof: First consider the claim that the integral is linear. It was shown above that theintegral is linear on Re

(L1 (Ω)

). Then letting a+ ib,c+ id be scalars and f ,g functions in

L1 (Ω) ,

(a+ ib) f +(c+ id)g = (a+ ib)(Re f + i Im f )+(c+ id)(Reg+ i Img)

= cRe(g)−b Im( f )−d Im(g)+aRe( f )+ i(bRe( f )+ c Im(g)+a Im( f )+d Re(g))

It follows from the definition that∫(a+ ib) f +(c+ id)gdµ =

∫(cRe(g)−b Im( f )−d Im(g)+aRe( f ))dµ

+i∫

(bRe( f )+ c Im(g)+a Im( f )+d Re(g)) (9.5)

Also, from the definition,

(a+ ib)∫

f dµ +(c+ id)∫

gdµ = (a+ ib)(∫

Re f dµ + i∫

Im f dµ

)+(c+ id)

(∫Regdµ + i

∫Imgdµ

)which equals

= a∫

Re f dµ−b∫

Im f dµ + ib∫

Re f dµ + ia∫

Im f dµ

+c∫

Regdµ−d∫

Imgdµ + id∫

Regdµ−d∫

Imgdµ.

Using Lemma 9.7.5 and collecting terms, it follows that this reduces to 9.5. Thus theintegral is linear as claimed.

Consider the claim about approximation with a simple function. Letting h equal anyof

(Re f )+ ,(Re f )− ,(Im f )+ ,(Im f )− , (9.6)

It follows from the monotone convergence theorem and Theorem 8.1.6 on Page 181 thereexists a nonnegative simple function s ≤ h such that

∫|h− s|dµ < ε

4 . Therefore, lettings1,s2,s3,s4 be such simple functions, approximating respectively the functions listed in9.6, and s≡ s1− s2 + i(s3− s4) ,∫

| f − s|dµ ≤∫ ∣∣(Re f )+− s1

∣∣dµ +∫ ∣∣(Re f )−− s2

∣∣dµ

+∫ ∣∣(Im f )+− s3

∣∣dµ +∫ ∣∣(Im f )−− s4

∣∣dµ < ε