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

Chapter 2

Introduction

This chapter contains an outline of a Riemann sum theory of random variability. Mathematical studies of random variation usually start with probability. In the traditional axiomatic approach these studies start with probability spaces (Ω, , P) where Ω is a sample space, is a sigma-algebra of measurable subsets of Ω, and P is a probability measure on Ω relative to .

In contrast, Chapter 1 introduces entities X ≃ x|Ω_X, F_X] and f(X) ≃ f(x)[Ω_X, F_X], called observables, as a basis for the Riemann sum approach of this book.

To begin with, for simplicity, we take a sample space Ω = [0, 1], the unit interval. In fact, though it is not always the simplest or most convenient choice of sample space, most random variables can be represented with [0, 1] as sample space. For example, in an experiment consisting of a single throw of a fair die (Example 8, Chapter 1), the sample space [0, 1] can replace {1, 2,...,6}, provided the potentiality distribution function is defined as follows:

with F_X(I) = 0 otherwise. Wiener [233], represented Brownian motion with [0, 1] as the sample ...