Bochner's theorem

For a locally compact Hausdorff Abelian group, a continuous positive-definite function $p$ is uniquely representable as $p(x)={\int }_{\hat{G}}\chi (x)d{\mu }_p(\chi )$ for a finite positive Radon measure on the Pontryagin dual group. Its total mass is $p(e)$. Conversely every such finite positive measure defines a continuous positive-definite function.