10.7. THE LEBESGUE INTEGRAL, L1 287

= 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)) (10.5)

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 10.7.5 and collecting terms, it follows that this reduces to 10.5. 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 )− , (10.6)

It follows from the monotone convergence theorem and Theorem 9.1.6 on Page 239 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 in10.6, 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

It is clear from the construction that |s| ≤ | f |.What about 10.4? 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 10.7.2. 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).