What is an integral?

Historically the concept of integration was first considered for real functions of a real variable where either the notion of ‘the process inverse to differentiation’ or the notion of ‘area under a curve’ was the starting point. In the first case a real number was obtained as the difference of two values of the ‘indefinite’ integral, while the second case corresponds immediately to the ‘definite’ integral. The so-called ‘fundamental theorem of the integral calculus’ provided the link between the two ideas. Our discussion of the operation of integration will start from the notion of a definite integral, though in the first instance the ‘interval’ over which the function is integrated will be the whole space. Thus, for ‘suitable’ functions f: Ω → R* we want to define the integral I(f) as a real number. The ‘suitable’ functions will be called integrable and I(f) will be called the integral of f.

Before defining such an operator I, we examine the sort of properties I should have before we would be justified in calling it an ‘integral’. Suppose then that A is a class of functions f: Ω → R*, and I:A → R defines a real number for every f∈A. Then we want I to satisfy:

1. (i) f∈A, f(x) ≥ 0 all x∈Ω ⇒ I(f) ≥ 0, that is I preserves positivity;

2. […]