Real Analysis/Generalized Integration

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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 $\mathbb {R}$ . 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 extention of the idea behind the Darboux integral

Upper and Lower sum

Let $f:[a,b]\to \mathbb {R}$

Let $\alpha :[a,b]\to \mathbb {R}$ such that $\alpha (x)$ is strictly increasing over $[a,b]$

Let ${\mathcal {P}}$ be a partition over $[a,b]$ , and let $x_{k},x_{k-1}\in {\mathcal {P}}$

The Upper Sum of $f$ with respect to ${\mathcal {P}}$ and $\alpha$ is given by

$U(f,{\mathcal {P}},\alpha )=\sum _{k=1}^{n}M_{k}{\big (}\alpha (x_{k})-\alpha (x_{k-1}){\big )}$

where $M_{k}$ is given as in the previous chapter.

The Lower Sum of $f$ with respect to ${\mathcal {P}}$ and $\alpha$ is given by

$L(f,{\mathcal {P}},\alpha )=\sum _{k=1}^{n}m_{k}{\big (}\alpha (x_{k})-\alpha (x_{k-1}){\big )}$

where $m_{k}$ is given as in the previous chapter.

Definition

Let $f:[a,b]\to \mathbb {R}$

Let $\alpha :[a,b]\to \mathbb {R}$ such that $\alpha (x)$ is strictly increasing over $[a,b]$

We say that $f$ is Riemann-Stieltjes integrable on $[a,b]$ with respect to $\alpha$ if and only if

$\sup _{\mathcal {P}}{\Big \{}L(f,{\mathcal {P}},\alpha ){\Big \}}=\inf _{\mathcal {P}}{\Big \{}U(f,{\mathcal {P}},\alpha ){\Big \}}$

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

$L=\sup _{\mathcal {P}}{\Big \{}L(f,{\mathcal {P}},\alpha ){\Big \}}=\inf _{\mathcal {P}}{\Big \{}U(f,{\mathcal {P}},\alpha ){\Big \}}$ is said to be the integral of $f$ on $[a,b]$ with respect to $\alpha$ and is denoted as $\int \limits _{a}^{b}f(x)d\alpha (x)$ or as $\int \limits _{a}^{b}fd\alpha$

Observe that putting $\alpha (x)=x$ , 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 $\delta :[a,b]\to \mathbb {R} ^{+}$ , that is, the range of $\delta (x)$ includes only positive reals.

A tagged partition ${\mathcal {\dot {P}}}={\Big \{}(t_{k},[x_{k-1},x_{k}]){\Big \}}_{k=1}^{n}$ is said to be δ-fine for a gauge $\delta$ if and only if for all $k$ ,

$[x_{k-1},x_{k}]\subseteq {\big (}t_{k}-\delta (t_{k}),t_{k}+\delta (t_{k}){\big )}$

Definition

Let $f:[a,b]\to \mathbb {R}$

Let $L\in \mathbb {R}$

Then, $f$ is said to be Henstock-Kurtzweil integrable on $[a,b]$ if and only if, for every $\varepsilon >0$ there exists a gauge $\delta :[a,b]\to \mathbb {R} ^{+}$ such that if ${\mathcal {\dot {P}}}$ is a δ-fine partition of $[a,b]$ , then

${\Big |}S(f,{\mathcal {\dot {P}}})-L{\Big |}<\varepsilon$