Real Analysis/Generalized Integration

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Real Analysis
Generalized Integration

The notion of integration is one of fundamental importance in advanced analysis. The idea of integration is expanded so as to be applicable to sets more general than subsets of . Interested readers may refer to the Wikibook Measure Theory. Here however, we will discuss two important generalizations of integration which are still applicable only to real valued functions.

Contents

Riemann-Stieltjes integralEdit

The Riemann-Stieltjes integral (or Stieltjes integral) can be seen as an extention of the idea behind the Darboux integral

Upper and Lower sumEdit

Let  

Let   such that   is strictly increasing over  

Let   be a partition over  , and let  

The Upper Sum of   with respect to   and   is given by

 , where   is given as in the previous chapter

The Lower Sum of   with respect to   and   is given by

 , where   is given as in the previous chapter

DefinitionEdit

Let  

Let   such that   is strictly increasing over  

We say that   is Riemann-Stieltjes integrable on   with respect to   if and only if

 , where the supremum and the infimum have been taken over the set of all partitions.

  is said to be the integral of   on   with respect to   and is denoted as   or as  

Observe that putting  , we get the Darboux integral, and hence, the Darboux integral is a special case of the Riemann-Stieltjes integral.

Henstock Kurtzweil integralEdit

While calculating the Riemann integral, the "fineness" of a partition was measured by it norm. However, it turns out that the norm is a very crude measure for a partition. Thus, by introducing the clever notion of gauges, we can extend the idea of the Riemann integral to a larger class of functions. In fact, it turns out that this integral, called the Henstock-Kurtzweil integral (after Ralph Henstock and Jaroslav Kurzweil) or Generalised Riemann integral is more general than the Riemann-Stieltjes integral and several other integrals on real intervals.

GaugesEdit

A Gauge is said to be a function  , that is, the range of   includes only positive reals

A tagged partition   is said to be δ-fine for a gauge   if and only if for all  ,


 

DefinitionEdit

Let  

Let  

Then,   is said to be Henstock-Kurtzweil integrable on   if and only if, for every   there exists a gauge   such that if   is a δ-fine partition of  , then