# Construction of the Wiener process and its continuity

Let $$C[0,1]$$ the space of continuous functions $$x:[0,1]\to \mathbb{R}$$ with $$x(0)=0$$. Equip $$C[0,1]$$ with the sigma algebra generated by the cylinder sets $$B_E=\{x\in C:(x(t_1),x(t_2),\dots,x(t_n))\in E\in B(\mathbb{R}^n)\}$$ which coincides with the Borel sigma algebra. Let $$W$$ denote the Wiener measure on $$B(C[0,1])$$ and consider the measure space $$(C[0,1],B(C[0,1]),W)$$. Then the functionals $$f_t:C[0,1]\to f_t(x)=x(t)$$ are normally distributed random variables $$N(0,t)$$. For $$t>s$$, $$f_t-f_s$$ is $$N(0,t-s)$$. And finally for $$t_3>t_2>t_1$$, $$f_{t_3}-f_{t_2}$$ is stochastically independent of $$f_{t_2}-f_{t_1}$$.

We see that $$(f_t)_{t\in[0,1]}$$ satisfies the properties of Brownian motion.

Is $$f_t(\omega)=B(t,\omega)$$ a Brownian motion for $$\omega\in C[0,1]$$? How do I show almost sure continuity?

Isn't it there in the very definition? For every fixed $$x \in C[0,1]$$ the sample path corresponding to it simply the function $$x$$ on $$[0,1]$$ which is continuous.
• Yes. The definition is $B(t,\omega)=\omega (t)$ for $\omega \in C[0,1]$ and $t \in [0,1]$. – Kabo Murphy Jul 18 at 9:23