This is an exercise from Prof. Amir Dembo's lecture notes for Stochastic Process course.
Consider a sequence of two coin tosses, $\Omega=\{HH,HT,TH,TT\}$, $\mathcal{F}=2^{\Omega}$, $\omega=(\omega_1 \omega_2)$,. Specify $\sigma(X_0)$, $\sigma(X_1)$, $\sigma(X_2)$ and $\sigma(X_i,i=0,1,2)$ for the R.V.-s:
$$X_0(\omega)=4\\ X_1(\omega)=2X_0(\omega)I_{\{\omega_1=H\}}(\omega)+0.5X_0(\omega)I_{\{\omega_1=T\}}(\omega),\\X_2(\omega)=2X_1(\omega)I_{\{\omega_2=H\}}(\omega)+0.5X_1(\omega)I_{\{\omega_2=T\}}(\omega).
$$
Now I also give the definition of $\sigma(X)$ here.
Definition: Given a R.V. X we denote by $\sigma(X)$ the smallest $\sigma$-field $\mathcal{G}\subseteq\mathcal{F}$ such that $X(\omega)$ is measurable on $(\Omega,\mathcal{G})$. One can show that $\sigma(X)=\sigma(\{\omega:X(\omega)\leq \alpha\})$. We call $\sigma(X)$ the $\sigma$-field generated by $X$.
I know the definition of $\sigma$-field and can understand how Borel $\sigma$-algebra is constructed. But I don't know how to specified a smallest $\sigma$-field for R.V.-s described above.
Best Answer
Hint:
$\sigma (\{ \omega : X (\omega) \le \alpha \})$ is the $\sigma$-algebra generated by the sets $\{ \omega : X (\omega) \le \alpha \}$ as $\alpha$ varies over $\mathbb R$.
For example, for $\alpha \ge 4$, $\{ \omega : X_0 (\omega) \le \alpha \} = \Omega$, whereas for $\alpha < 4$, $\{ \omega : X_0 (\omega) \le \alpha \} = \emptyset$. Hence $\sigma(X_0) = \sigma(\{ \Omega, \emptyset\})$. (Which $\sigma$-algebra is this equal to?)
Can you take it from here? Note that since your sample space is finite, you can construct $\sigma$-algebras using finite unions, intersections, and complements.