For a bounded state space $[a,b]$ and a non-negative operator $T_K$ with kernel $K$, on $L_2([a,b])$, Mercer's theorem yields that $T_K$ is trace class if and only if
$$\int_a^{b} K(x, x) dx = \sum_{i} \lambda_i = \operatorname{trace}{(T_k)}\,.$$
Is there a version of this for when the state space is unbounded, say $\mathbb{R}^d$? That is, for a (Markov) operator defined on $L_2( \pi)$ where $\pi$ has unbounded support, can we say that $T_K$ is trace-class if and only if $$\int K(x, x) dx < \infty\,, $$ and $$\operatorname{trace}(T_k) = \int K(x, x) dx? $$
General results would be fine, and a result specific to Markov operators would also be sufficient.