= Solution
\b[Yes.] Let $A$ be $T$-invariant. Since the itinerary variables generate $\mathcal A$ modulo null sets, $A$ can be regarded as an event in the canonical sequence model. For every $N$, invariance and $X_n\circ T=X_{n+1}$ imply
$$
A=T^{-N}A\in\sigma(X_N,X_{N+1},\ldots).
$$
\b[Thus $A$ belongs to the <tail sigma-algebra> of the i.i.d. sequence $(X_n)$. By the <Kolmogorov zero-one law>, $m(A)$ is zero or one. Hence the <tent map> is an <ergodic measure-preserving transformation>.]
Back to article page
