I am working through Handbook of Heavy-Tailed Distributions in Asset Management and Risk Management by Bianchi et al. At the very beginning of chapter three it dives into stochastic processes and definitions for them based on the pair $(\Omega, \mathcal{F})$ that brings a universe and a $\sigma$-algebra. Where I am stuck is as follows:
A stochastic process is a collection of random variables $X = \{X_t\}_{t\geq 0}$ on $(\Omega, \mathcal{F})$ which can take values on $(E, \mathcal{E}$). A process may be considered a mapping from $(\Omega \times \mathbb{R}_+, \mathcal{B}(\mathbb{R}_+) \otimes \mathcal{F})$ into $(E, \mathcal{E})$ (usually $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$), via $(\omega, t) \mapsto X(\omega, t) = X_t(\omega)$.
(Emphasis mine.). I have gone through examples of filtrations, e.g. this one with coin tosses. I am seeking to fit the coin toss problem into the emphasized part of the definition given above.
In my thinking, $\Omega$ is the universe; $\mathbb{R}_+$ is, in the coin example, a subset of $\mathbb{N}$ for indexing; $\mathcal{F}$ is the filtration based on the history of coin flips (so $\mathcal{F} = (\mathcal{F_n})$); and the $\mathbb{R}$ in the image of the process is again $\mathbb{N}$ for indexing the coin toss sequence. I am stuck, however, on 1) if this is correct, 2) what is the Borel $\sigma$-algebra in the second component of the domain of the coin tosses, and 3) what is the Borel $\sigma$-algebra over $\mathbb{R}$ in the range? How would all of this fit into the coin toss example?
Thank you.
