{A direct construction of the Wiener measure on $\textbf{C}[0, \infty)$

TL;DR

This paper presents a direct construction method for the Wiener measure on the space of continuous functions over [0, ∞), using a set function based on normal distributions and a structural relation, bypassing the traditional process-based approach.

Contribution

It introduces a novel direct construction of the Wiener measure on continuous function spaces, avoiding the usual Brownian motion process formulation.

Findings

Constructed Wiener measure directly on C[0, ∞)

Used a set function based on n-dimensional normal distributions

Established measure on the Borel σ-field of C[0, ∞)

Abstract

Our construction of the Wiener measure on $\textbf{C}=\textbf{C}[0, \infty)$ consists in first defining a set function $\varphi$\ on the class of all compact sets based on certain $n$-dimensional normal distributions, $n = 1,\ 2,\ldots$\ using the structural relation at (\ref{E1.2}) below. This structural relation, discovered by the first author, is recorded in his book (2013) on page 130. We then define a measure $\mu$ on the Borel $\sigma$-field of subsets of $\textbf{C}$ which is the Wiener measure. This is done via a similar construction of the Wiener measure on $\textbf{C}_a=\textbf{C}[0, a)$ where $a > 0$ is an arbitrary real number. The traditional way is to first construct the Brownian Motion process (BMP) and then, by proving it is a measurable mapping into $(\textbf{C},\ \mathscr{C}_\infty)$, call the measure induced by the BMP on $\textbf{C}$\ the Wiener measure. In the present paper, we define the Wiener measure directly.