From: "Oleg V. Poliannikov" <ovpolian@eos.ncsu.edu>
Subject: Re: Riemann integration
Date: Sun, 19 Sep 1999 00:01:41 -0400
Newsgroups: sci.math
Keywords: Standard definition of Riemann integral

> > Not even the limit of a sequence, since uncountably many objects are
> > taken into account.
>
> You need to go back to your textbook to understand the definition of a
> Riemann integral.

Thank you very much for the advise. I won't use if you don't mind since
I know the definition by heart.

I'll give it to you so that you would also learn something.

Let [a,b] be a segment.

Definition.
We will call T a partition of [a,b] if T={x_n},n=0...N, a=x_0<...<x_N=b.

Definition.
The diameter of a partition (d(T)) is a value max_{i=1...N}{x_i-x_{i-1}}

Definition.
We will call a partition T a marked partition if it is endowed with
the sequence \ksi_n,n=1...N such that
\forall i=1...N x_i-1 < \ksi_i < x_i.

We will denote the set of all marked partiotions as P

Note! (For those who advise others to read books)
P is uncountable.

Let f:[a,b]->R

Definition.
A Riemann sum of function f is mapping from P to R defined by the following
formula:
S_f(T)=\sum_{i=1}^{N}{f(\ksi_i)(x_i-x_{i-1})}

Definition.
A function f is called Riemann integrable is there exists a limit
J = lim_{d(T)->0}{S_f(T)}

J is then called the Riemann integral of f over [a,b].



Hope it wasn't very difficult for you.

Oleg*