I'm trying to figure out the meaning of the following theorem from J. Norris - Markov Chains.
Let $$\gamma_i^k=E_k\sum_{n=0}^{T_k-1}1_{\{X_n=i\}}$$ Let $P$ be irreducible and let $\lambda$ be an invariant measure for $P$ with $\lambda_k=1$.
Then $\lambda \ge \gamma^k$. If in addition $P$ is recurrent, then $\lambda=\gamma^k$.
The way I see it, this means that if there's an invariant measure for which $\lambda_k=\gamma_k^k=1$, then this invariant measure $\lambda$ is definitely greater or equal to the $\gamma^k$ invariant measure.
I'm not looking for a proof, but rather an understanding of why the following theorem holds true. What's the intuitive way to think about it?
Thanks in advance!