Chapter 1

The Riemann Integral

The Riemann integral is a fundamental part of calculus and an essential

precursor to the Lebesgue integral. In this chapter we define the Riemann

integral of a bounded function on an interval I — [a, b] on the real line.

To do this, we partition i" into smaller intervals. A partition V of / is a

finite collection of subintervals {J/~ : 0 k N}, disjoint except for their

endpoints, whose union is /. We can order the Jk so that Jk — [xk,Xk+i],

where

(1.1) xo x\ • • • XN XN+1, #o = a, XN+I = b.

We call the points x^ the endpoints of V. We set

(1.2)

We then set

(1.3)

t{Jk) —

xk+i

—

xk,

maxsize (V) = max £(Jk),

0kN

minsizefP)^ min £(Jk)-

0kN

k

Jk

/p(/) = £inf/0rK(Jfc).

Note that Lp(f) ^ Iv(f) These quantities should approximate the Riemann

integral of /, if the partition V is sufficiently "fine."

To be more precise, if V and Q are two partitions of /, we say V refines

Q, and we write V - Q, if V is formed by partitioning each interval in Q.

Equivalently, V - Q if and only if all the endpoints of Q are also endpoints

1

http://dx.doi.org/10.1090/gsm/076/01