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

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