From the paper: “G. Takeuti introduced the Fourier transform for mappings defined on a locally compact Abelian group and having as values commuting normal operators in a Hilbert space. The aim of the present article is to develop G. Takeuti’s results in the following directions: First, we consider more general arrival spaces, namely, order-complete lattice-normed spaces. Thereby, we cover the following two important particular cases: Banach spaces and \(K\)-spaces. Second, we introduce the class of dominated mappings and demonstrate that the class coincides with the set of all inverse Fourier transforms of quasi-Radon vector measures with bounded variation. Last, in our approach we eliminate the construction of a Boolean-valued universe from the definitions and statements of theorems.
We denote by \(qca(X,Y)\) the lattice-normed space of all \(\sigma\)-additive quasi-Radon measures on \(B(X)\) with values in \(Y\). The norming \(K\)-space of \(qca(X,Y)\) is the space \(qca(X,F)\) of all \(F\)-valued \(\sigma\)-additive quasi-Radon measures. The main result of the article is the following Theorem. Given a mapping \(\varphi:G \to Y\), the following assertions are equivalent: (1) \(\varphi\) has dominant \(o\)-continuous at zero; (2) there exists a unique measure \(\mu\in qca(X,Y)\) such that \[ \varphi (g)=\int_X \chi(g) \mu(d\chi) \quad g\in G \text{''}. \]