By Prentice R.L.

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**Example text**

But how far can we go in neglecting wild points? What is the largest collection of wild points that can be safely ignored in integration? The answer to this question is furnished by Lebesgue's theory of sets of points of zero measure. 2. 1. A set of points Eon the real axis is said to have zero one-dimensional measure if, to each positive number E there corresponds an enumerable collection of intervals {In} (n = 1, 2, ... ), which cover the set E and whose total length DEFINITION does not exceed E.

Then f(x) < f(s") if s < x < s". (2) Sincef(x) is continuous, there is an interval a< x < sin which f(x) < f(s"). Let s' be the greatest lower bound of the points a at the lower extremity of these intervals. Then s'

LP The limit functionf(x) has the value unity if xis rational and the value zero if x is irrational. The upper and lower Darboux integrals are respectively I and 0, whence f(x) has no Riemann integral. To be just to the Riemann integral, which has played so great a part in nineteenth-century analysis, it must be stated that it does possess a certain restricted continuity, in the sense that if the functions {fn{x)} are each integrable by Riemann's method, if the sequence {fn{x)} converges uniformly to the limit function f(x), and iff(x) is also integrable by Riemann's method, then b b J fn(x) dx ~ J f(x) dx a as n ~ oo.