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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 {\displaystyle [ a,b ] } be a given interval. By a partition {\displaystyle P } of {\displaystyle [ a,b ] } we mean a finite set of points {\displaystyle x_0,x_1,\cdots,x_n, } where
                {\displaystyle a = x_0 \le x_1 \le \cdots \le x_{n-1} \le x_n = b }.
We write
              {\displaystyle \Delta x_i = x_i - x_{i-1} \;\;\; (i=1,\ldots,n) }.
Now suppose {\displaystyle f } is a bounded real function defined on {\displaystyle [ a ,b ] }. Corresponding to each partition {\displaystyle P } of {\displaystyle [ a,b ] } we put
                {\displaystyle M_i = \sup f(x) \;\;\;\;\; (x_{i-1}\le x \le x_i) },
                {\displaystyle m_i = \inf f(x) \;\;\;\;\;\;\ (x_{i-1}\le x \le x_i) },

                {\displaystyle U(P,f) = \sum_{i=1}^n M_i \Delta x_i },
                {\displaystyle L(P,f) = \sum_{i=1}^n m_i \Delta x_i },

and finally

(1)                {\displaystyle \overline{\int_a^b} f \ dx = \inf \ U(P,f) },
(2)                {\displaystyle \underline{\int_a^b} f \ dx = \sup \ L(P,f) },

where the {\displaystyle \inf } and {\displaystyle \sup } are taken over all partition {\displaystyle P } of {\displaystyle [ a,b ] }. The left members of (1) and (2) are called the upper and lower Riemann integrals of {\displaystyle f } over {\displaystyle [ a,b ] }, respectively.

If the upper and lower integrals are equal, we say that {\displaystyle f } is Riemann-integrable on {\displaystyle [ a , b ] }, we write {\displaystyle f \in \mathscr{R} } (that is, {\displaystyle \mathscr{R} } denotes the set of Riemann-integrable functions), and we denote the common value of (1) and (2) by

(3)                {\displaystyle \int_a^b f \ dx },

or by

(4)                {\displaystyle \int_a^b f(x) \ dx }.

This is the Riemann integral of {\displaystyle f } over {\displaystyle [ a,b ] }. Since {\displaystyle f } is bounded, there exist two numbers, {\displaystyle m } and {\displaystyle M }, such that

                    {\displaystyle m \le f(x) \le M \;\;\;\;\;\; (a \le x \le b) }.

Hence, for every {\displaystyle P },

                {\displaystyle m(b-a) \le L(P,f) \le U(P,f) \le M(b-a) },

so that the numbers {\displaystyle L(P,f) } and {\displaystyle U(P,f) } form a bounded set. This shows that the upper and lower integrals are defined for every bounded function {\displaystyle f }

 

注意点は,リーマン積分の定義は被積分関数 {\displaystyle f } に有界な実数値関数であることを要請している点でしょうか.また,上積分と下積分が一致すればリーマン可積分でありリーマン積分が存在する,ということですが,そのためは {\displaystyle f } に(有界な実数値関数であることに加えてさらに)どのような要請が必要であるか,をこの定義から知ることはできないようです.

以上,引用しただけですが,リーマン積分の定義を調べました.


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