Group-metric spaces and a class of \(p\)-adic valued measures (Q2783756)

Let \(G\) be a topological group acting isometrically on a metric space \((X,d)\). The author introduces the concept of a balanced p. b. compact subset of a complete metric space. The main result of this paper (Theorem 3.4) establishes a criterion for a continuous function \(f: C\to {\mathbb C} _p\) to be Riemann-integrable, where \(C\) is balanced p. b. compact in \({\mathbb C} _p\), with respect to a certain \(p\)-adic valued bounded measure \(\mu _C\). In fact \(\mu _C\) is a probability function. As a consequence we have that, in general, the indentity function is not Riemann integrable. Therefore this result gives a partial answer to a problem of \textit{V. Alexandru, N. Popescu} and \textit{A. Zaharescu }[J. Number Theory 88, 13-38 (2001; Zbl 0965.11049); ibid. 68, 131-150 (1998; Zbl 0901.11035)]. NEWLINENEWLINENEWLINEThis paper gives a general framework for problems considered by several authors before. The organization of the paper is as follows. First several results relative to \(G\)-metric spaces, the definition of p. b. compact set and a natural generalization of the classical Krasner's Lemma, are given. Next, the author constructs the \(p\)-adic measure \(\mu _C\). In the last section the ring of continuous functions on a p. b. compact set is studied and the main result proved.