# Chapter 4

# Theory of the Integral

In this chapter, using Definition 15 of the Burkill-complete integral in Chapter 3, theorems needed for study of random variability are stated and proved. The basic features and structure of the Henstock integral have been introduced in preceding chapters and are formally presented here.

## 4.1 The Henstock Integral

A system, called a *division system*^{1} is posited, with the following components and relationships.

- There is a domain
*W*, consisting of points*x*, with a class**I**(*W*) of subsets, called*cells*of*W*. The domain*W*belongs to**I**(*W*). - The union
*E*of a finite number of cells*I*is called a*figure*. The class of figures in*W*is denoted by**E**(*W*). - A finite collection of disjoint cells
*I*whose union is*E*is called a*partition*of*E*(often denoted ).

**DS1 Association Axiom:** For each *I* **I**(*W*) there is a class of points *I**, and if *x* *I** the pair (*x*, *I*) are said to be *associated.*

- Equivalently, for each
*x**W** there is a class*x** of cells*I*, and if*I**x** the pair (*x*,*I*

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