According to http://www.randomservices.org/random/processes/Stop.html, a stochastic process $X=\{X_t:t \in T\}$ is a stochastic process with state space $(S, \mathscr{S})$ defined on underlying probability space $(\Omega, \mathscr{F}, P)$.
Suppose we have a simple stochastic process $U=\{U_1, U_2, U_3..\}$ defined on $\Omega$ = {H, T}, which is toss a coin, $\mathscr{F}$ = $=\{\emptyset, \{H\}, \{T\}, \{H, T\}\}$, $P(H) = p$, $P(T) = 1-p$, and for any $i = 1,2...$, we have $U_i(H) = 1, U_i(T) = -1$, which means state space $S$ = {1, -1} and $\mathscr{S} = \{\emptyset, \{1\}, \{-1\}, \{1, -1\}\}$.
Now we calculate $\sigma\{U_1\}$, $\sigma\{U_1\} = \sigma\{U_1^{-1}(A): A \in \mathscr{S}\} = \{\emptyset, \{H\}, \{T\}, \{H, T\}\}$, simialarly, $\sigma\{U_1, U_2\} = \sigma\{U_1^{-1}(A), U_2^{-1}(A): A \in \mathscr{S}\} = \{\emptyset, \{H\}, \{T\}, \{H, T\}\}$, but intuitively for me, this seems wrong!
So where did I make mistakes?