This argument is taken from Resnicks Adventures in stochastic processes and let $T _{\infty } < \infty $ denote that an infinite number of transitions in a continuous time markov chain has occurd in finite time. [We have constructed the continuous time markov chain from a markov chain with descrete time .] I hope I do not have to exactly define what is meant with $T _{\infty } < \infty $, or how the markov chain is contructed, but that the result don't hinge on this.
Let $$P \left[T_\infty < \infty \mid \{X _n \}\right]=1_{\left[\sum \frac {1 } {\lambda (X _n ) } \right]} \text{ a.s.} $$
Then resnick says " Taking expectation yields"
$$P\left[T _\infty < \infty |X (o) =i \right]=P \left[\sum \frac {1 } {\lambda (X _n ) } |X (o) =i \right],$$
where $\lambda (i ) >0$ for every state $i $.
I don't understund what have been used here.
Thanks in advance!