Mercer's theorem

Let $X$ be a compact metric space, let $\mu$ be a finite Borel measure whose support of a measure is all of $X$, and let $K$ be a continuous Hermitian positive-semidefinite kernel. The integral operatoris a positive compact operator on $L^2(\mu )$. Its strictly positive eigenvalues ${\lambda }_j$ have orthonormal eigenfunctions with continuous representatives ${\varphi }_j$, and Mercer's theorem givesThe series converges uniformly on $X\times X$ and absolutely at every pair of points. These measure and support assumptions specify where the eigenfunction expansion reconstructs the original kernel.^[ref]

For why full support of a measure matters, take $X=\{0,1\}$, $\mu ={\delta }_0$, and $K(1,1)=1$ with all other values zero. This continuous positive-semidefinite kernel gives the zero integral operator on $L^2(\mu )$, so its eigenfunctions cannot recover the value at the unmeasured point1. For a full-support measure, integration of the diagonal expansion instead gives ${\sum }_j{\lambda }_j={\int }_{X}K(x,x)d\mu (x)$. This connects the trace-class operator trace to the diagonal of its continuous integral kernel.