Let $(X, \mathcal{A}, m, T)$ be a probability measure preserving system.

(a) State what it means for $(X, \mathcal{A}, m, T)$ to be ergodic.

(b) State Kolmogorov's 0-1 law for a sequence of independent random variables. What does it imply for the canonical model associated with an i.i.d. random process?

(c) Consider the special case when $X=[0,1], \mathcal{A}$ is the $\sigma$-algebra of Borel subsets, and $T$ is the map defined as

$$T x=\left\{\begin{array}{l} 2 x, \quad \text { if } x \in\left[0, \frac{1}{2}\right] \\ 2-2 x, \quad \text { if } x \in\left[\frac{1}{2}, 1\right] \end{array}\right.$$

(i) Check that the Lebesgue measure $m$ on $[0,1]$ is indeed an invariant probability measure for $T$.

(ii) Let $X_{0}:=1_{\left(0, \frac{1}{2}\right)}$ and $X_{n}:=X_{0} \circ T^{n}$ for $n \geqslant 1$. Show that $\left(X_{n}\right)_{n \geqslant 0}$ forms a sequence of i.i.d. random variables on $(X, \mathcal{A}, m)$, and that the $\sigma$-algebra $\sigma\left(X_{0}, X_{1}, \ldots\right)$ is all of $\mathcal{A}$. [Hint: check first that for any integer $n \geqslant 0, T^{-n}\left(0, \frac{1}{2}\right)$ is a disjoint union of $2^{n}$ intervals of length $1 / 2^{n+1}$.]

(iii) Is $(X, \mathcal{A}, m, T)$ ergodic? Justify your answer.