Hyperfinite dimensional representations of spaces and algebras of measures (Q2505583)

This paper is a contribution to the general theory of local approximation of topological groups resp. algebras by (hyper)finite groups resp. algebras.\ This theory has been completely renewed by the use of nonstandard methods. More specifically, the authors focus on the hyperfinite representation of spaces of measures, following, for example, previous works of P. A. Loeb. The framework is the following: \(\mathbf{X}\) denotes a locally compact topological space, \((X,E,X_{\omega})\) any triple formed by a hyperfinite set \(X\) in a sufficiently saturated nonstandard universe, a monadic equivalence relation \(E\) on \(X\) and an \(E\)-closed galactic set \(X_{\omega}\subset X\) such that all internal subsets of \(X_{\omega}\) are relatively compact in the induced topology and \(\mathbf{X}\) is homeomorphic to the quotient \(X_{\omega}/E\). The authors show that each regular complex Borel measure on \(X\) can be obtained by pushing down the Loeb measure induced by some internal function \(X\rightarrow{^{\ast}\mathbb{C}}\). Moreover, the construction gives rise to an isometric isomorphism of the Banach space \(M(\mathbf{X})\) of all regular complex Borel measures on \(\mathbf{X}\), normed by total variation and the quotient \(\mathcal{M} _{\omega}(X) /\mathcal{M}_0(X)\), for some external subspaces \(\mathcal{M}_{\omega}(X) \) and \(\mathcal{M}_0(X)\) of the hyperfinite-dimension\-al space \(^{\ast}\mathbb{C}^{X}\) with the norm \(\| f\| _1=\sum_{x\in X}| f(x)|\). Finally, if \(X\) is a hyperfinite group, \(X_{\omega}\) is a galactic subgroup of \(X\), \(E\) the equivalence corresponding to a normal monadic subgroup \(X_0\) of \(X_{\omega}\) and \(\mathbf{X}\) is isomorphic to the locally compact group \(X_{\omega}/X_0\), then the above Banach space isomorphism turns to be an isomorphism of Banach algebras.