Traceability of positive integral operators in the absence of a metric

We investigate the traceability of positive integral operators on $L^2(X,\mu )$ when $X$ is a Hausdorff locally compact second countable space and $\mu$ is a non-degenerate, $\sigma$-finite and locally finite Borel measure. This setting includes other cases proved in the literature, for instance the one in which $X$ is a compact metric space and $\mu$ is a special finite measure. The results apply to spheres, tori and other relevant subsets of the usual space ${\mathbb{R}}^m$.