= Solution
Let $\nu$ be an <invariant distribution> for the <continuous-time Markov chain>. Using $Q=D_g(P-I)$, its <invariant distribution of a continuous-time Markov chain>[stationarity equation] $\nu Q=0$ becomes
$$
(\nu D_g)P=\nu D_g.
$$
Thus the measure with weights $\nu_i g(i)$ is invariant for the irreducible positive-recurrent <jump chain>. Its <invariant distribution> $\pi$ is unique, so $\nu_i g(i)=c\pi_i$. Normalization gives the <invariant-measure transfer between a jump chain and a CTMC> formula
$$
\boxed{\nu_i=\frac{\pi_i/g(i)}{\displaystyle\sum_{j\in S}\pi_j/g(j)}}.
$$
The denominator is finite and nonzero because $\varepsilon<g(j)<1/\varepsilon$. Thus $\nu$ is a <probability distribution> and $(X_t)$ is positive recurrent.
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