Sample space and $\sigma$-algebra for a stochastic process

Question

I am trying to get a grasp on the choice of $\sigma$-algebra and sample space for stochastic processes. Given a sequence of random variables $\left( X_t\right)_t$, we need a probability space $(\Omega, \mathcal{F}, P)$ and a filtration $(\mathcal{F_t})_t$, such that for every $t$, $X_t$ is $\mathcal{F}_t$-measurable.

I wonder though, how $\mathcal{F}$ and $\Omega$ may be constructed. I begin with an example.

I consider the process of tossing a fair coin. If I knew how many tosses I would perform, let's say three, I would have $\Omega = \{ H, T \}\times \{ H, T \}\times \{ H, T \}$. However, when I am faced with a process $(X_t)_t, \, t\in \mathbb{N}$ it becomes confusing. Does it make sense to take $\Omega = \{H, T\}^\mathbb{N}$? Does such an object even exist?

Answer 1

The step from finite case to infinite case is not so obvious and requires the very famous Kolmogorov extension theorem.