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

Appendix A

Epilogue

A.1   Measurability

Without invoking a theory of measure, Chapter 4 demonstrates that Riemann sums, constructed from point/cell/dimension-set association and regulated by gauges, deliver a theory of integration which has the characteristics, properties, and theorems required for the purposes of real analysis, and in particular for the purposes of analysis of random variability.

Sections 4.7 and 4.8 of Chapter 4 introduced variation as a form of outer measure of subsets S of R^T. In Section 4.9 this concept was extended from sets S in R^T to sets S × in R^T × , and the extended concept of outer measure was used in Theorem 52. This in turn was neded when establishing the integrability of functions that arise in quantum mechanics—in Theorem 219, for instance.

The Henstock or Burkill-complete integral of a function h of elements x, N, I[N], structured by a multifunctional relation of association between these elements, is defined by forming Riemann sums of the function values, the terms of the Riemann sums being selected by “gauge” rules. Thus the concept of measurability is not required in order to define the integral. However, in proving certain properties of the integral, or in calculating the integrals of particular functions, the analytical device of step functions ...