= Solution
A distribution $\pi$ is an <invariant distribution> when
$$
\pi_j=\sum_{i\in I}\pi_i p_{ij}
$$
for every state $j$. The pair $(\pi,P)$ satisfies <detailed balance> when
$$
\pi_i p_{ij}=\pi_jp_{ji}
$$
for every $i,j$. Summing this identity over $i$ gives
$$
\sum_i\pi_i p_{ij}
=\sum_i\pi_jp_{ji}
=\pi_j\sum_i p_{ji}
=\pi_j,
$$
so detailed balance implies invariance.
For an irreducible positive recurrent <Markov chain>, <Kac's lemma> relates the invariant mass to the <mean recurrence time>:
$$
\boxed{\mathbb E_iT_i^+=\frac1{\pi_i}}.
$$
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