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

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?