Fourier transforms of finitely additive bounded measures Given a finitely additive positive regular bounded measure $\mu$ on ${\mathbb R}^n$ (i.e. a positive linear functional on $C_b({\mathbb R}^n)$), I wonder what can be said about its Fourier transform. In case of a countably additive measure, Bochner's theorem tells us that the Fourier transform is a continuous positive definite function. This makes use of dominated convergence, which - I assume - fails in general in the finitely additive case. So I wonder how/if such measures can be characterized in terms of their Fourier transforms.
 A: I think there is an error in the statement of the question. $C_b(\mathbb{R}^n)$ is isomorphic, as an ordered Banach space, to $C(\beta \mathbb{R}^n )$ where $\beta \mathbb{R}^n$ is the Stone Cech compactification of $\mathbb{R}^n$. By Riesz's theorem, for example Chapter 7 in Folland's "Real Analysis: modern techniques and their applications", the dual of $C(\beta \mathbb{R}^n)$ is the space of finite $\sigma$-additive, signed Radon measures $M(\beta \mathbb{R}^n)$. The positive part of the space of globally defined functionals over $C_b(\mathbb{R}^n)$, by the closed graph theorem, equals the positive cone of $M(\beta \mathbb{R}^n)$. In order to describe the finitely additive measures you need to work with the (positive part of the) dual of $L^{\infty}(\mathbb{R}^n)$.
That said, your question admits a partial answer. The Fourier transform of any
finitely additive measure will be a (discontinuous) positive type function 
over $\mathbb{R}_{disc}^n$. For the converse only the existence holds, i.e.: for any positive type function $\phi$ over $\mathbb{R}_{disc}^n$ there are infinitely many elements in $L^{\infty}(\mathbb{R}^n)^{\ast}_{+}$ whose Fourier transform is $\phi$.
Let $\chi_{\theta}(x) = e^{ \theta x}$ be a character and let $m \in L^{\infty}(\mathbb{R}^n)^{\ast}$. We define the Fourier transform of $m$, as $\mathcal{F}m(\theta) = m(\chi_{\theta})$. As the function $\theta \to \chi_{\theta}$ is not continuous all we can say is that $\mathcal{F}m(\theta) \in C_b(\mathbb{R}_{disc}^n)$ and that $\| \mathcal{F}m(\theta) \|_{\infty} \leq \|m\|_{L^{\infty}(\mathbb{R}^n)^{\ast}}$. Notice that when $m$ is positive $\| m \|_{L^{\infty}(\mathbb{R}^n)^{\ast}} = m(\mathbb{1}_{\mathbb{R}^n})$. If $a_1, a_2, ... a_k \in \mathbb{C}$ and $\theta_1, \theta_2, ... \theta_k \in \mathbb{R}^n$ then:
$$ \sum_{i,j}{ \bar{a_i} a_j \mathcal{F}m(\theta^{-1}_i \theta_j) } = 
   m( ( \sum_{ i }{ \bar{a_i} \bar{ \chi_{ \theta_i } } } )
      ( \sum_{ j }{ a_j \chi_{ \theta_j } } ) ) \geq
   0
$$
To see that the injectivity fails let $m \in L^{\infty}(\mathbb{R}^n)^{\ast}_{+}$ be a translation invariant and unital element, which exist because $\mathbb{R}^n$ is amenable. The translation invariance implies that $\mathcal{F}m(\theta) = \chi_{\eta} \mathcal{F}m(\theta)$ for every $\eta$. So $\mathcal{F}m$ can either be identically 0 or be $\mathbb{1}_{\{0\}}$. Since $m$ is unital we are in the second case. As there are infinitely many invariant means on an amenable group, see exercise 14 in Paterson's Amenability, the difference of any two will be in the kernel of the Fourier transform.
The surjectivity can be proved by Bochnner's Theorem in LCA groups applied to 
the Bohr compactification of $\mathbb{R}^n$. There is a surjective, order preserving map $\pi$ from $L^{\infty}(\mathbb{R})^{\ast}$ to the Borel, $\sigma$ -additive measures in the Bohr compactification $M(b \mathbb{R}^n)$ given by dualizing the injective inclusion of $C(b \mathbb{R}^n)$ into $L^{\infty}(\mathbb{R}^n)$ obtained by restricting any continuous function in $b \mathbb{R}^n$ to the open dense subset $\mathbb{R}^n \subset b \mathbb{R}^n$. Bochnner's theorem for $b \mathbb{R}^n$ will give you that any positive type function on $\mathbb{R}_{disc}^n$ is given by the Fourier transform of a positive measure $\mu$ in $b \mathbb{R}^n$. Taking any element in $\pi^{-1}[\{\mu\}]$ suffices to see the surjectivity.
I hope this is Helpful for you.
