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In this definition of Wiener's measure, we define the measure of a standard Brownian motion by extending the f.d.d. distribution on set of all continuous functions. However, we also know that the set of all continuous functions is not measurable in the measurable space of all functions, with $\sigma$-field generated by all pointwise projection maps $\pi_t(\omega)=\omega(t)$. This fact, in some textbooks(like Durrett's PTE), asks us to constraint functions on rational points and then show that the sample paths are uniformly continuous with probability 1.

So my question is that whether my following understanding is correct:

  1. if we start with all functions $\{\omega:[0,\infty)\mapsto\mathbb{R}\}$ then the measurability of $C(\mathbb{R})$ is indeed a problem, which requires a detour via rational approximations;

  2. but if we start with $C(\mathbb{R})$ then everything is fine. But in this case what is the $\sigma$-field?

  3. so the reason for starting from all functions to construct the Brownian motion is that we want more generality, that we don't want to assume in advance that the sample paths are continuous almost surely?

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Your understanding is correct. If you wanted to start with $C=C(\mathbb R_+\to\mathbb R^n)$ (the space of all continuous functions from $\mathbb R_+$ to $\mathbb R^n$) then the $\sigma$-algebra is $C\cap {\cal B}^{\mathbb R_+}$ which is the collection of all sets of the form $C\cap B$ where $B$ is in the $\sigma$-algebra ${\cal B}^{\mathbb R_+}\,.$ This in turn is the $\sigma$-algebra on the space of all maps $\mathbb R_+\to\mathbb R^n$ (which can be denoted by $(\mathbb R^n)^{\mathbb R_+}$) such that every projection $\mathbb R_+\ni t\mapsto \omega(t)\in \mathbb R^n$ is Borel measurable.

To make a long story short: the space of all maps and the $\sigma$-algebra ${\cal B}^{\mathbb R_+}$ are not easy to avoid when one starts to construct the Wiener measure with finite dimensional distributions (Kolmogorov theorem).

As far as I remember, the alternative Levy-Ciesielski construction builds more directly on continuous functions.

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