13.1. DEFINITIONS AND BASIC PROPERTIES 305

Proof: Let |α| = 1 and α∫

I f dF = |∫

I f dF |. Then by Theorem 13.1.9 and Lemma13.1.10, ∣∣∣∣∫I

f dF∣∣∣∣ =

∫Iα f dF =

∫I(Re(α f )+ i Im(α f ))dF

=∫

IRe(α f )dF + i

∫IIm(α f )dF

=∫

IRe(α f )dF ≤

∫I| f |dF

Note the assumption that | f | ∈ R∗ [a,b]. I will point out later that you can’t assume | f |is also generalized Riemann integrable. This is just like the case with series. A series mayconverge without converging absolutely.

The following lemma is also fundamental. It is about restricting f to a smaller intervaland concluding that the function is still generalized Riemann integrable on this smallerinterval.

Lemma 13.1.12 If f ∈ R∗ [a,b] and [c,d]⊆ [a,b], then f ∈ R∗ [c,d].

Proof: Let ε > 0 and choose a gauge δ such that if P is a division of [a,b] which isδ fine, then |S (P, f )−R| < ε/2. Now pick a δ fine division of [c,d] ,{(Ii, ti)}l

i=r and let{(Ii, ti)}r−1

i=1 , {(Ii, ti)}ni=l+1 be fixed δ fine divisions on [a,c] and [d,b] respectively.

Now let P1 and Q1 be δ fine divisions of [c,d] and let P and Q be the respective δ finedivisions of [a,b] just described which are obtained from P1 and Q1 by adding in {(Ii, ti)}r−1

i=1and {(Ii, ti)}n

i=l+1. Then

ε > |S (P, f )−R|+ |S (Q, f )−R| ≥ |S (Q, f )−S (P, f )|= |S (Q1, f )−S (P1, f )| .

By the above Cauchy criterion, Proposition 13.1.5, f ∈ R∗ [c,d] as claimed.

Corollary 13.1.13 Suppose c ∈ [a,b] and that f ∈ R∗ [a,b] . Then f ∈ R∗ [a,c] and f ∈R∗ [c,b]. Furthermore, ∫

If dF =

∫ c

af dF +

∫ b

cf dF.

Here∫ c

a f dF means∫[a,c] f dF.

Proof: Let ε > 0. Let δ 1 be a gauge function on [a,c] such that whenever P1 is a δ 1fine division of [a,c], ∣∣∣∣∫ c

af dF−S (P1, f )

∣∣∣∣< ε/3.

Let δ 2 be a gauge function on [c,b] such that whenever P2 is a δ 2 fine division of [c,b],∣∣∣∣∫ b

cf dF−S (P2, f )

∣∣∣∣< ε/3.

Let δ 3 be a gauge function on [a,b] such that if P is a δ 3 fine division of [a,b] ,∣∣∣∣∫ b

af dF−S (P, f )

∣∣∣∣< ε/3.