Degrees of GeneralityDate: 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/ |
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