I wish to define the Wiener process and some associated diffusions. For this, I thought to work with the space $C([0,\infty),\mathbb{R})$ of continuous functions, endowed with the sup-norm $||f||=\sup(|f(s)|:s\geq 0)$, $f\in C([0,\infty),\mathbb{R})$. The norm induces a metric and topology and as I understand, this induces the Borel $\sigma$-field generated by the open sets. However, in the book Brownian motion and siffusion by David Freedman, the space is indeed $C([0,\infty),\mathbb{R})$ but what he calls the Borel $\sigma$-field is $\sigma(X(t),t\geq 0)$. It's tough for me to tell if these definitions are the same or not, I have a feeling they are not, but I'm not sure. To me, Freedman's definition sounds not well defined because he doesn't define the topology, so why use the name Borel. Is his $\sigma$-field the Borel $\sigma$-field that corresponds to the discrete topology on $C([0,\infty),\mathbb{R})$?
As you can see, I'm quite confused even in these basic definitions. Thanks for your help.
The topology of uniform convergence on $C(ℝ_+, ℝ)$ is not the one used in the context of the Wiener process, since it is too fine. The definition in your book, $\mathcal C := σ(X(t), t \geq 0)$, refers to the σ-algebra on $C(ℝ_+, ℝ)$ generated by the canonical projections $$ X(t): C(ℝ_+, ℝ) \ni w ↦ w(t) ∈ ℝ, $$ which happens to coincide with the restriction of the product-σ-algebra $\mathcal B(ℝ)^{⊗ℝ_+}$ to $C(ℝ_+,ℝ)$. In particular $\mathcal C$ contains all the cylinder sets in $C(ℝ_+, ℝ)$. It is indeed the case that $\mathcal C$ is generated by a topology on $C(ℝ_+, ℝ)$, so it is really a Borel-σ-algebra. This generator is not the uniform topology, but rather the topology of compact convergence (i.e. uniform convergence on all compact subsets). The proof of this is sketched in this answer.