My financial mathematics course notes have
A Brownian motion is a family of random variables $\{B_t|t\geq0\}$ on some probability space $(\Omega,\mathcal{F},P)$ such that: \begin{align} (1) \; & B_0=0, \\ (2) \; & \text{for } 0\leq s\leq t \text{ the increment } B_t-B_s \text{ is normally distributed} \\ & \text{with mean } 0 \text{ and variance } t-s, \\ (3) \; & \text{for any } 0\leq t_1<t_2<\dots<t_n \text{ the increments } \\ & \hspace{1em} B_{t_1}-B_0,B_{t_2}-B_{t_1},\dots,B_{t_n}-B_{t_{n-1}} \\ & \text{are independent random variables, and} \\ (4) \; & \text{For any } \omega \in \Omega \text{ the function } t\mapsto B_t(\omega) \text{ is continuous.} \end{align}
I interpret the numbered $\omega$'s in the images (low res in the original) to mean that a Wiener process/Brownian motion is typically (for this context) taken over a fixed $\omega$. What I then wonder is: where does the randomness come from?
To try to understand better, I found these notes, which have
The space of elementary events for the Brownian motion is the set of all continuous real functions, $$\Omega=\{\omega(t):\mathbb{R}_+\mapsto\mathbb{R}\}.$$
and
Random functions: For each $t\geq0$ consider the random variable $X_t(\omega)=\omega(t)\in\Omega$. This random variable is the outcome of the experiment of sampling the position of a Brownian particle (trajectory) at a fixed time $t$. Thus $X_t(\omega)$ takes different values on different trajectories.
If $\omega$ is any continuous, real-valued function, shouldn't a sequence of such experiments on a fixed $\omega$ sometimes produce constant-valued curves?