# リーマン積分の定義を調べる

リーマン積分(Riemann integral)とはなにか？という問いに答えられるようにするため，リーマン積分の定義を調べることにしました．

Rudin(1976)から引用します．

6.1 Definition   Let be a given interval. By a partition of we mean a finite set of points where
.
We write
.
Now suppose is a bounded real function defined on . Corresponding to each partition of we put
,
,

,
,

and finally

(1)                ,
(2)                ,

where the and are taken over all partition of . The left members of (1) and (2) are called the upper and lower Riemann integrals of over , respectively.

If the upper and lower integrals are equal, we say that is Riemann-integrable on , we write (that is, denotes the set of Riemann-integrable functions), and we denote the common value of (1) and (2) by

(3)                ,

or by

(4)                .

This is the Riemann integral of over . Since is bounded, there exist two numbers, and , such that

.

Hence, for every ,

,

so that the numbers and form a bounded set. This shows that the upper and lower integrals are defined for every bounded function

[1] Rudin, W. (1976), Principles of Mathmatical Analysis (Third Edition), McGraw-Hill,Inc.