I read the following text in one of the book on Random Processes, but I am not able to understand the complete meaning of it.
Consider a sample space ${\{H,T}\}^{\mathbb N}$. We define a mapping $X:\Omega \to\{0,1\}^ \mathbb N$ such that $X_n(\omega) = \mathbb{1}_{\{H\}}(\omega_n)$. The map X is a $F$-measurable random sequence, if each $X_n : \Omega \to \{0,1\}$ is a bi-variate $F$-measurable random variable on the probablity space $(\Omega,F,P)$. Therefore, the event space must contain the event space generated by events $E_n = \{\omega \in \Omega : X_n(\omega)=1\}$ i.e. $\sigma(X) = \sigma(E_n:n \in \mathbb N)$.
I know that the event space generated by a random process is give as $\sigma(X)=\sigma(A_{X{_t}}(x):t \in T, x \in R)$ where $A_{X_t}(x) = {X_t}^{-1}(-\infty , x]$. But it is not clear to me how they have generated their event space in this case. Also I am not aware what are bi-variate $F-$measurable random variables.
Any hint or suggested reading in this regard will be very helpful. Thanks!
Let $A=\{H,T\}$. Note that $ X_n^{-1}((-\infty, x])=\underbrace{A\times \cdots \times A}_{n-1}\times\{T\}\times A\times\cdots=E_n^c $ for any $x<1$, and $X_n^{-1}((-\infty, x])=A^{\mathbb{N}}$ for all other $x$'s.
So, according to your definition, $$ \sigma(X)=\sigma(E_n^c:n\in \mathbb{N})=\sigma(E_n:n\in \mathbb{N}) $$ because if $E_n^c\in\sigma(X)$, then both $E_n$ and $E_n\cup E_n^c=A^{\mathbb{N}}$ belong to $\sigma(X)$.