Solution
= Solution
Sum the identity from part (a) over $z\in A$. Since $X_{T_A}\in A$ almost surely under positive recurrence,
$$
1=\sum_{z\in A}\pi(z)\mathbb E_zT_A^+.
$$
Using the conditional stationary law $\pi_A(z)=\pi(z)/\pi(A)$ gives
$$
\mathbb E_{\pi_A}T_A^+
=\frac1{\pi(A)}
\sum_{z\in A}\pi(z)\mathbb E_zT_A^+.
$$
Hence the <Kac lemma>[return-time identity]
$$
\boxed{\mathbb E_{\pi_A}T_A^+=\frac1{\pi(A)}}.
$$