Theorem 5 (Bochner)

A continuous function $\varphi$ on $\widehat{G}$ belongs to Pos($\widehat{G}$) if and only if there exists a bounded positive measure $\nu$ on G such that $\varphi =\mathscr{F}(\nu )$.

Let $\nu \in \mathscr{M}^1(G)$ be a positive measure. Its Fourier transform is continuous. For every finite family $(x_i)_{i\in I}$ in $\widehat{G}$ and every finite family $(t_i)_{i\in I}$ of complex numbers, one obtains

since $\nu$ is a positive measure. The Fourier transform of $\nu$ is therefore a function of positive type on $\widehat{G}$ (Th. 1 of V, p. 432, (ii)).

Let us prove the converse. Endow the set Pos$_{\leqslant 1}(\widehat{G})$ with the weak topology, induced as above by the weak topology $\sigma (L^{\infty}(\widehat{G}),L^1(\widehat{G}))$. The set Pos$_{\leqslant 1}(\widehat{G})$ is compact and convex (Cor. to Prop. 13 of V, p. 448). By Prop. 14 of V, p. 450, Prop. 11 of V, p. 444 and Cor. 7 of V, p. 390, the extreme points of Pos$_{\leqslant 1}(\widehat{G})$ are the zero function and the elements of $\widehat{G}$.

Let $\mathscr{N}$ be the set of bounded positive measures of mass $\leqslant 1$ on G; it is compact in $\mathscr{M}^1(G)$ endowed with the weak topology (EVT, III, p. 17, Cor. 3). By Lemma 9 and the first part of the proof, the Fourier transform on G defines, by passing to subspaces, a continuous mapping of $\mathscr{N}$ into Pos$_{\leqslant 1}(\widehat{G})$; by homogeneity, it is enough to prove that this mapping is surjective. The image $\mathscr{F}(\mathscr{N})$ of $\mathscr{N}$ under the Fourier transform is convex and compact; it contains the zero function and the elements of $\widehat{G}$ (in fact, these are of the form ev$_x:\chi \mapsto \chi (x)$ for an element $x$ of G by Th. 2 of II, p. 220, and one has ev$_x=\mathscr{F}(\varepsilon_{x^{-1}})$). The set $\mathscr{F}(\mathscr{N})$ therefore contains the extreme points of Pos$_{\leqslant 1}$(G), whence $\mathscr{F}(\mathscr{N}) =$ Pos$_{\leqslant 1}(G)$ by the Krein-Milman theorem (EVT, II, p. 59, Th. 1). This concludes the proof.

When $G =\mathbf{R}^k$ for an integer $k\in \mathbf{N}$, this theorem corresponds to Prop. 11 of INT, IX, p. 94, § 6, n$^o12$.