# Real Analysis/Generalized Integration

←Darboux Integral | Real Analysis Generalized Integration |
Pointwise Convergence→ |

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[edit]

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

### Upper and Lower sum[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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