1549901369-Elements_of_Real_Analysis__Denlinger_

7.3 The Integral as a Limit of Riemann Sums 383

Theorem 7.3.15 (Regular Partition Riemann/Darboux Criterion for

Integrability) A bounded function f :[a, b] --7 JR is integrable over [a, b] {::=:?

I Ve> 0, 3no EN 3 n :'.'.'.no==? S(f, Qn) - S..(f, Qn) < e.1

Proof. Exercise 19. •

Continuing along this line, the limit criterion for integrability comes out in
the following form using regular partitions.

Theorem 7.3.16 (Regular Partition Limit Criterion for Integrability)

A bounded f :[a, b] --7 JR is integrable over [a, b] and J: f = I {::=:?

Ve > 0 , 3 no E N 3 n :'.'.'. no ==? JR(!, Q~) - II < e.

(regardless of the choice of the tags xi)

Proof. Exercise 20. •

Finally, because of Theorem 7.3.16, it makes sense to write

n

t a f = n-+oo lim R(f, Q~) = n-+oo lim b-a n '"""f ~ (xi)

i=l

if this limit exists and is independent of the tags xi.

It must be emphasized that this "limit" is not the ordinary limit of a
sequence, as defined in Chapter 2. It is a special kind of limit, whose definition
is given in the statement boxed in Theorem 7.3.16. Instead of a statement
about one sequence, it is a statement about all possible sequences {R(f, Q~)}
generated by different selections of the tags xi. There are infinitely many such
sequences.
Encouraged by the news that regular partitions are sufficient, the reader
might also naively hope that in the Riemann sum approach, the common prac-
tice of using left endpoints, right endpoints, or midpoints as the tags xi is
sufficient. The following example dispels that hope.

Example 7.3.17 Consider the Dirichlet function,

f(x) = { 1 if xis rational }

0 if x is irrational

on the interval [O, l]. If Q is any regular partition of [O, 1], then all Riemann
sums using left endpoints, right endpoints, or midpoints of the subintervals