13.1. DEFINITIONS AND BASIC PROPERTIES 307

Proof: Let (Jk, tk) ∈ P\P′. From Lemma 13.1.12, f ∈ R∗ [Jk]. Therefore, letting Qk bea suitable δ fine division of Jk, using δ as a generic gauge as small as the original δ ,∣∣∣∣∫Jk

f dF−S (Qk, f )∣∣∣∣< η

|P\P′|+1

where η > 0 and |P\P′| denotes the number of intervals from P which are not in P′. Thereare |P\P′| different values of k. Let P̃ be the division which results from all the Qk alongwith P′. We modify the original δ only on the intervals of P\P′ always making it smaller.Then

ε >

∣∣∣∣S(P̃, f)−∫

If dF

∣∣∣∣=

∣∣∣∣∣∣(

r

∑j=1

f(ti j

)∆Fi−

r

∑j=1

∫Ii j

f dF

)+

|P\P′|∑k=1

S (Qk, f )−∫

Jk

f dF

∣∣∣∣∣∣≥

∣∣∣∣∣ r

∑j=1

f(ti j

)∆Fi−

r

∑j=1

∫Ii j

f dF

∣∣∣∣∣− η |P\P′||P\P′|+1

>

∣∣∣∣∣ r

∑j=1

f(ti j

)∆Fi−

r

∑j=1

∫Ii j

f dF

∣∣∣∣∣−η

Then∣∣∣∑r

j=1 f(ti j

)∆Fi−∑

rj=1∫

Ii jf dF

∣∣∣< ε +η and since η is arbitrary,

∣∣∣∣∣ r

∑j=1

f(ti j

)∆Fi−

r

∑j=1

∫Ii j

f dF

∣∣∣∣∣≤ ε

Consider{(I j, t j)

}pj=1 a subset of a division of [a,b]. If δ is a gauge and

{(I j, t j)

}pj=1 is

δ fine, meaning that I j ⊆ (t j−δ (t j) , t j +δ (t j)) , this can always be considered as a subsetof a δ fine division of the whole interval and so the following corollary is just a rewordingof the above.

Lemma 13.1.15 Suppose that f ∈ R∗ [a,b] and that whenever Q is a δ fine division ofI = [a,b], ∣∣∣∣S (Q, f )−

∫I

f dF∣∣∣∣≤ ε.

Then if {(Ii, ti)}pi=1 is δ fine, maybe not dividing all of I, it follows that∣∣∣∣∣ p

∑j=1

f (t j)∆F (I j)−p

∑j=1

∫I j

f dF

∣∣∣∣∣≤ ε.

Here is another corollary in the special case where f has real values. In this lemma,one splits the indices into those for which f (ti)∆Fi ≥

∫Ii f dF and the ones in which the

inequality is turned around. You can do this because of the above corollary.