How can we identify $\omega\in\Omega$ with a path of Brownian motion $t\rightarrow B_t(\omega)$? In the Stochastic Differential Euqations written by Oksendal(see page 12),

As we shall soon see, the paths of a Brownian motion are (or, more correctly, can be chosen to be) continuous, a.s. Therefore we may identify (a.a.) ω ∈ Ω with a continuous function $t → B_t(ω)$ from $[0, \infty)$ into $\mathbb{R}^n$. Thus we may adopt the point of view that Brownian motion is just the space $C([0,\infty),\mathbb{R}^n)$ equipped with certain probability measures $P_x$ (given by (2.2.1) and (2.2.2) above).

I have no idea about identification of $\omega$ with a continuous function $t\rightarrow B_t(\omega)$. 
I will be so happy for someone to explain this concept..
How can we identify $\omega\in\Omega$ with a continuous function $t\rightarrow B_t(\omega)$?
Thanks in advance!
 A: Suppose you have a bernoulli variable. Since $X(\omega)$ is either $1$ or $0$, you can identify $\omega$ with the value that $X(\omega)$ takes on (with of course $P^X$, the measure induced by $X$ on $\Omega$)
We do this all the time; when you say that $X \sim N(0,1)$ you're not specifying $\Omega$ or the exact $X$, but you just say that $X(\omega) \in \mathbb R$ and you give a certain probability $P^X$ for each interval of $\mathbb R$. In a certain sense you identified $\omega$ with it's image under $X$, that is $X(\omega)$ and you gave a probability distribution on those values. In this sense, instead of specifying the probability space $(\Omega, \mathcal A, P)$ and the random variable $X :\Omega \mapsto \mathbb R$ you are specifying the "transformed" probability space ($\mathbb R, \mathcal B(\mathbb R), P^X)$. You can then identify $\Omega$ with $\mathbb R$ under the "transformation" applied by $X$. 
A brownian motion is a stochastic process, a collection of random variables; so instead of having a single value for every $\omega$, you have an infinite amount. Since for every fixed $\omega$ you find a trajectory $t \mapsto B_t(\omega)$ which is (almost surely) a continuous function, you can identify $\omega$ with it's image under the map $\omega \mapsto B_\cdot (\omega)$ where $B_\cdot (\omega)$ is a continuous function; this map then associates to a certain $\omega$ a certain continuous function, so you identify $\Omega$ with the space of all continuous function from $[0, \infty)$ to $\mathbb R^n$. So in analogy with the previous case, you don't specify $(\Omega, \mathcal A, P)$ and $B_t(\omega)$ for every $t$, but you just specify the "transformed" space $C([0, \infty), \mathbb R^n)$ with a suitable sigma algebra and a suitable probability measure.
