I use the following definition for a stochastic process.
Let $(\Omega, \mathcal F, P)$ be a probability space, $(E, \mathcal E)$ be a measurable space, and $T$ be a non-empty set. A collection $X:=(X_t)_{t\in T}$ of random variables is called a stochastic process with state space $(E, \mathcal E)$ indexed by $T$ and defined on $\Omega$ if for all $t \in T$, $X_t: \Omega \to E$ is $\mathcal F/\mathcal E$ measurable. In addition, for $\omega \in \Omega$ the function $t \mapsto X_t(\omega)$ is called the sample path of $X$.
My question is how to interpret the $\omega$ in $X_t(\omega)$ in the above definition. Consider the infinite fair coin tossing experiment. In this case, it is clear that one sample path can be $$X_t(\omega) = 0101011101010 \cdots, {t \in \{1, 2, 3, \cdots\}}.$$ What exactly is the $\omega$ in this sample path, please? How do I understand it? Thank you!