Let X be a nonempty measurable subset of \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{R}^m$$\end{document} and consider the restriction of the usual Lebesgue measure σ of \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{R}^m$$\end{document} to X. Under the assumption that the intersection of X with every open ball of \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{R}^m$$\end{document} has positive measure, we find necessary and sufficient conditions on a L2(X)-positive definite kernel \documentclass[12pt]{minimal}
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\begin{document}$$K : X \times X \rightarrow \mathbb{C}$$\end{document} in order that the associated integral operator \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {K} : L^2(X) \rightarrow L^2(X)$$\end{document} be nuclear. Taken nuclearity for granted, formulas for the trace of the operator are derived. Some of the results are re-analyzed when K is just an element of \documentclass[12pt]{minimal}
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\begin{document}$$L^2(X \times X)$$\end{document}.