A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration

Chapter 3

Infinite-Dimensional Integration

The previous chapter gives an overview of the Henstock approach to integration in progressively more “demanding” domains of integration. This requires characterization of the point-cell pairs (x, I) of the various domains; and of various forms of linkage or association between the points and cells used in Riemann sums; and of various rules for selection of associated (x, I) in Riemann sums.

In the basic Riemann-complete integral the selection rule involves a positive function δ(x), often called a gauge. The δ-gauges method in Riemann-complete integration is an instance of a system or set of rules for determining variable elements in Riemann sums, a system that finds full expression in the Henstock integral, as set out in Chapter 4. We will apply the term gauge as a shorthand description of any such set of rules, once the variable elements and the relationships between them have been decided upon. Then, depending on the gauge chosen, the Henstock integral reduces to the Riemann, Lebesgue, Riemann-complete (Henstock-Kurzweil) and other integrals.

Having established, firstly, the basic framework of associated point-cell pairs, and, secondly, gauges (or rules) for selecting the terms to be used in Riemann sums, the actual definition of the Henstock integral of a function depends on a straightforward inequality involving Riemann sums, as in Definition 10 in Chapter 2.

These, then, are the basic building blocks of the theory. There are no special ...