Degrees of Generality

Date: 03/13/99 at 00:17:30
From: Pete
Subject: Different types of Integrals

In my calculus class, I learned the Riemann integral. However, in my 
side readings, I came across terms like Riemann-Stieltjes integral and 
even Lebesgue integral. They seem to be different; could you explain 
why we need so many different integrals? 

Thank you.

Date: 03/13/99 at 08:29:26
From: Doctor Jerry
Subject: Re: Different types of Integrals

These integrals represent degrees of generality. For a function to be 
Riemann integrable its domain of definition must be something like a 
ray or an interval and on its domain, it must be continuous except on 
a set of measure zero.

The Riemann-Stieltjes integral manages to combine the Riemann integral 
and infinite series into one idea. Statisticians like this integral.

The Lebesgue integral or its generalizations to "measure theory" are 
designed to handle more general functions and to help out with the 
interchange of integration and various limit operations.  

For example, suppose that for each n, f_n(x)  (f_n means f sub n) is a 
continuous function on an interval [a,b]. Is it true that for Riemann 
integrals,

lim_{n->oo} int(a,b,f_n(x)*dx)=int(a,b,lim_{n->oo} f_n(x)*dx) ?

This is false for Riemann integrals. For example, let a = 0, b = 1, and 
define f_n to be the function whose graph is the line seg from (0, 0) 
to (1/(2n), n) and then the line segment from (1/(2n), n) to (1/n, 0), 
and then the line segment from (1/n, 0) to (1, 0). We see that 
int(0, 1, f_n(x)*dx) = 1/2 and so the left side is 1/2. But the point-
wise limit lim_{n->oo} f_n(x) = 0 for all x in [0, 1]. Hence the right 
integral is 0.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/