= Mercer's theorem
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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 operator>
$$
(Tf)(x)=\int_XK(x,y)f(y)\,d\mu(y)
$$
is a positive <compact operator> on $L^2(\mu)$. Its strictly positive <eigenvalues> $\lambda_j$ have <orthonormal> <eigenfunctions> with continuous representatives $\phi_j$, and <Mercer's theorem> gives
$$
K(x,y)=\sum_j\lambda_j\phi_j(x)\overline{\phi_j(y)}.
$$
The <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>.<What is an RKHS? by Dino Sejdinovic and Arthur Gretton>[section6.2, Theorem48, printed p21]{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 $\sum_j\lambda_j=\int_XK(x,x)\,d\mu(x)$. This connects the <trace-class operator> trace to the diagonal of its continuous <integral kernel>.
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