I am studying continuous time Markov chains on a countable state space and my professor said the following: A state $i$ is positive recurrent if and only if $$\liminf_{t\to\infty} \frac{1}{t} \int_0^t P_{i,i}(s)\textit{d}s>0$$ where P is the semigroup associated to the Markov chain. I do not understand why this is true. Might someone have some insights?
Thank you in advance!