Real Analysis/Generalized Integration

../ Generalized Integration Darboux Integral Pointwise Convergence __TOC__ 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. Riemann-Stieltjes integral The Riemann-Stieltjes integral (or Stieltjes integral) can be seen as an extension of the idea behind the Darboux integral Upper and Lower sum 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. Definition 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 integral 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. Gauges 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 , Definition 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