I am currently working through a text which uses Bochner's Theorem. It cites its statement as follows.
Let $G$ be a locally compact abelian group, and let $\widehat{G}$ be its dual group. Then there is a bijection between
- continuous positive definite functions $f\colon G\to \mathbb{C}$, and
- nonnegative bounded Radon measures $\mu$ on the Borel sets of $\widehat{G}$
given by the Fourier transform $$f(x) = \int_\widehat{G} \chi(x)\mathrm{d}\mu(\chi).$$
It then proceeds to make use of $\mu$ being a Radon measure by using that the union of all open null sets is also a null set (which, as far as I understand, doesn't hold for more general measures).
However, on Wikipedia and in other sources, Bochner's theorem is stated for probability measures instead of Radon measures:
Let $G$ be a locally compact abelian group, and let $\widehat{G}$ be its dual group. Then there is a bijection between
- continuous positive definite functions $f\colon G\to \mathbb{C}$ with $f(0)=1$, and
- probability measures on the Borel sets of $\widehat{G}$
given again by the Fourier transform as above.
From my understanding, these two statements contradict each other since not every probability measure on a locally compact abelian group is a Radon measure. So my question is: Is one of these false, or am I misunderstanding something?