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],
(1.1) xo x\ XN XN+1, #o = a, XN+I = b.
We call the points x^ the endpoints of V. We set
We then set

maxsize (V) = max £(Jk),
minsizefP)^ min £(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
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