The material on Stieltjes integrals has a very important generalization called integration on
rough functions. This chapter gives an introduction to this topic. In order to show this,
we need a simple inequality called the triangle inequality. First here is a useful
lemma.

As in the case of Stieltjes integrals all of this has generalizations to integrator functions
which have values in various normed linear spaces but this is a book on single variable
advanced calculus and so this level of generality is avoided. However, it is just a matter
regarding F has having values in a vector space. All the details are the same other than
this.

Lemma 10.0.1If a,b ≥ 0 and p^{′}is defined by

1
p

+

1
p′

= 1, then

ap bp′
ab ≤ p + p′ .

Proof: Let p^{′} = q to save on notation and consider the following picture:

PICT

∫ a ∫ b p q
ab ≤ tp−1dt+ xq−1dx = a-+ b-.
0 0 p q

Note equality occurs when a^{p} = b^{q}. ■

The following is a case of Holder’s inequality.

Lemma 10.0.2Let a_{i},b_{i}≥ 0. Then

( )1∕p( )1∕p′
∑ ab ≤ ∑ ap ∑ bp′
i ii i i i i

Proof:From the above inequality,

∑n ---ai--------bi-----
(∑ ap)1∕p(∑ p′)1∕p′
i=1 i i ibi

( ) ( ′ )
∑n 1 -api-- 1- --bpi--
≤ p ∑i api + p′ ∑ bp′
i=1( ∑ p ) (∑ pi′)i
= 1 ∑-iai- + 1- --ibi- = 1 + 1-= 1
p iapi p′ ∑i bpi′ p p′

Hence the inequality follows from multiplying both sides by

(∑ ap)
i i

^{1∕p}

( )
∑ bp′
i i

^{1∕p′
}.■

Then with this lemma, here is the triangle inequality.