An elementary construction of the Wiener measure

TL;DR

The paper presents a straightforward construction of the Wiener measure on continuous functions by defining a set function on compact sets using normal distributions and extending it to the Borel sigma-algebra.

Contribution

It introduces a simple, elementary method for constructing the Wiener measure based on a structural relation involving normal distributions.

Findings

Constructed Wiener measure on continuous functions

Utilized a structural relation for defining set functions

Provided a measure consistent with classical Wiener measure

Abstract

Our construction of the Wiener measure on $\mathfrak{C}$ 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 (1) below. This structural relation, discovered by the first author, is recorded in his book [2] on page 130. We then define a measure $\mu$ on the Borel $\sigma$-field of subsets of $\textbf{C}$ which is the Wiener measure.