A sample path of a stochastic process $x:\Omega\times T\to\mathbb{R}^n$ is defined as taking $x(\omega,t)$ with fixed $\omega\in\Omega$, thus $x(\omega,\cdot)$.
However I have the following issue with this definition: the process is supposed to be a sequence of random variables $X_t:\Omega\to\mathbb{R}^n$, so a sample path, or realization would be a realization for each of the random variables in the sequence, leading to (maybe) different values of $\omega\in\Omega$, one for each $t\in T$.
For example if $\Omega = \{H,T\}$ and $T=\{0,1,\dots\}$ and all random variables $X_t$ are independent with $P(X_t = H) = p\ \ \forall t\in T$, then a realization would be something like $\{H,T,T,H,\dots\}$.
But, what am supposed to interpret by "fixing $\omega\in\Omega$", since one would only get sequences like $\{H,H,H,\dots\}$.
I've seen other answers here, and some say that you should not fix $\omega$ for all random variables in the sequence, or that $\omega$ is fixed on sequences in $\Omega^T$ (then $\omega\in\Omega^T$?). Of course that seems reasonable, but it doesn't follow the definition.
Am I missing something?
(Im using "Introduction to Stochastic Control" by Åström, Karl Johan and "Stochastic Differential Equations" by Øksendal, Bernt)