Is there a version of Bochner's theorem characterizing measures of positive type on a locally compact group? By a measure of positive type on the group $\Gamma$, I mean a measure $\mu$ satisfying $\int f^\* \* f~d\mu \geq 0$ for all $f \in C\_c(\Gamma)$. If so, what is a reference?


I am mainly interested in the case when $\Gamma$ is a compact group.



Bochner's theorem for measures of positive type
On $\Gamma = \mathbf R^n$, there is a Bochner theorem for not only measures but *distributions of positive type*, in Schwartz, Théorie des distributions, Chap. VII, §9:



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> Théorème XVIII (Bochner) Pour qu'une distribution $T$ soit $>>0$, il faut et il suffit qu'elle soit transformée de Fourier d'une mesure $\mu\geqslant0$ à croissance lente.
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For general locally compact groups, there is a theory of distributions by [Bruhat](http://ams.org/mathscinet-getitem?mr=140941), but I don't know if anyone proved a Bochner theorem in that context.


If you only want *measures*, there is a Bochner theorem on locally compact *abelian* groups, by [Argabright and Gil de Lamadrid](http://ams.org/mathscinet-getitem?mr=621876):



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> THEOREM 4.1. Every positive definite measure on $\Gamma$ is transformable. A measure $\mu\in\mathfrak M\_T(\Gamma)$ is positive definite if and only if $\hat\mu\geqslant0$.
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(Here they call a mesure $\mu$ on $\Gamma$ *transformable*, $\mu\in\mathfrak M\_T(\Gamma)$, if there is a measure $\hat\mu$ on the Pontryagin dual group $G$ such that $\mu(f^\*\*f) = \hat\mu(|\check f|^2)$ for all compactly supported continuous functions on $\Gamma$.)


Finally if you insist on *non-abelian* $\Gamma$, Argabright and Gil de Lamadrid point to a generalization by [Godement (1957)](http://www.numdam.org/item?id=SB_1956-1958__4__95_0)(p.101) where it is called "Théorème de Plancherel". (I haven't read it.) 


