 Solution

Write $Z(u)$ for the continuous-time Markov chain and ${\mathbb{E}}_r$ for expected value conditional on $Z(0)=r$. The occupation time of a continuous-time Markov chain in state $s$ is the integral of its indicator random variable. By Tonelli theorem,The interchange is valid because the integrand is nonnegative; indeed the random integral is bounded by $t$.