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

Abstract

Our construction of the Wiener measure on $\text{C}=\text{C}[0,\infty )$ consists in first defining a set function $\phi$\ on the class of all compact sets based on certain $n$-dimensional normal distributions, $n=1,2,\dots$\ 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 $\text{C}$ which is the Wiener measure. This is done via a similar construction of the Wiener measure on ${\text{C}}_a=\text{C}[0,a)$ 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 $(\text{C},{\mathcal{C}}_{\infty })$, call the measure induced by the BMP on $\text{C}$\ the Wiener measure. In the present paper, we define the Wiener measure directly.

16 pages. arXiv admin note: text overlap with arXiv:2011.05584