The Generalized Riemann Integral

Abstract

Riemann integration theory integrates functions on a bounded interval  as a Riemann sum approach (integral) where the fineness of the partitions is controlled by a number (norm) of the partition. In Generalized Riemann integral theory, the Riemann sum approach of functions is controlled by a gauge on tagged partition so that enabling integrating functions with much larger collections. Therefore, the theorems that apply to Generalized Riemann integral theory have differences in their hypotheses and conclusions. In this paper, theory of Generalized Riemann integral is studied by giving some examples of functions that are Generalized Riemann integrable such that they are not Riemann integrable; and proving some theorems that apply in this theory. The functions are integrable by constructing a gauge on the tagged partition of the interval such that the Riemann sum of the function is very close to some real number. Functions defined on a bounded interval that are Generalized Riemann integrable such that they are or not Riemann integrable have the general form of the function: a function f on [a,b] is continuous on [a,b]\Z and discontinuous on Z, where Z is a null set. Moreover, an unbounded function f on [a,b] is integrable, if the set Z where f is unbounded on Z is a countable set. Furthermore, these two criteria can be extended to infinite intervals, i.e. a function defined on an infinite interval can be Generalized Riemann integrable such that it is not Riemann integrable, if the set of discontinuous and unbounded points of the function is a null set. A sequence of integrable functions on an interval I that converges to a function on I, satisfies that this limit function is integrable if it satisfies that the existence of the dominating functions.