Characterization of integrals with respect to arbitrary Radon measures by the boundedness indices (Q1930230)

Let \(T\) be a set, and denote by \(F( T)\) the family of all functions \(f:T\rightarrow \mathbb{R}\). Let \(A( T) \subset F( T)\) be a lattice linear space of functions on \(T\) which is truncatable or has the Stone property, i.e., the condition \(f\in A( T)\) implies \(f\wedge \mathbf{1}\in A(T)\). The main result of the paper is a parametric theorem on the functional description of all Radon integrals. Theorem 5. Let \((T,\mathcal{G})\) be a Hausdorff space and \(A(T)\) be a truncatable lattice linear subspace in the space \(S(T,\mathcal{G})\) possessing the property \((E_{\sigma })\) or the property \((E)\) \((D)\). Suppose that \(\varphi \in A( T) ^{\Delta}\). Then the following assertions are equivalent: (1) the functional \(\varphi \) is natural; (2) there exists a (unique) Radon measure \(\mu \) such that \( E\mu =J\varphi \). Moreover, \(\varphi f=\int fd\mu \) for all \(f\in A(T)\) and the bijection \(I:\varphi \rightarrow \mu \) of \((A(T) ^{\Delta}) _{\mathrm{nat}}\) onto \(\mathcal{\Re M}(T,\mathcal{G},A(T))\) preserves all linear and lattice structures inherited by the family of functionals and the family of measures from their lattice linear spans. For notations, see the paper.