1549901369-Elements_of_Real_Analysis__Denlinger_

7.2 The Riemann Integral Defined 361

a b x

(a) (b)

Figure 7.1

For i = 1, 2, · · · , n , the ith subint erval of P is [Xi-l, Xi], and we define [See

Figure 7.1 (b).]

mi= inf f[xi- 1, xi] =inf {f(x ) : x E [xi-1,xi]};

Mi = sup f[xi-1, xi]= sup {f(x ) : x E [xi-1, xi]};

L::::.i =Xi - Xi-l = length of the ith subinterval [ xi-l, xi].

For each partit ion P we define the upper and lower Darboux sums,

n

(the lower Darboux sum for f over P).

i=l

n

S(f, P) = L Mif::::.i (the upper Darboux sum for f over P).

i= l

a= Xo X 1 X2 X3 X4 X5 x6 X7 xg = b a = Xo X j X2 X3 X4 X5 x6 X7

l (j. P) S(J. P)

Figure 7.2

Lemma 7.2.2 For all partitions P of [a, b], $_(!, P) ~ S(f, P).

Proof. Exercise 1. •

Definition 7 .2.3 A partition Q of [a, b] is a refinement of a partition P of
[a, b] if P ~ Q.