Integrable functions for Bernoulli measures of rank 1. II (Q2853427)

In [Ann. Math. Blaise Pascal 17, No. 2, 341--356 (2010; Zbl 1207.26031)] and the Ph.D. Thesis [Measures \(p\)-adiques et suites clasiques de nombres, Université de Bamako (2011)], the author began the study of non-Archimedean integrable functions with respect to the normalized Bernoulli measures of rank \(1\), \(\mu_{1,\alpha}\), where \(\alpha\) is a \(p\)-adic unit different from \(1\). The present paper explains the results of these two works. Let \(K\) be a complete valued field extension of the field \(\mathbb{Q}_p\) of the \(p\)-adic numbers (\(p\) a prime number) and let \(\mathbb{Z}_p\) be the ring of \(p\)-adic integers. Following the non-Archimedean integration theory developed by Monna and Springer in the 1960's, the author identifies the functions \(\mathbb{Z}_p \rightarrow K\) that are integrable with respect to the measures \(\mu_{1,\alpha}\). The main achievements of the paper are the following (Theorems 1, 2).NEWLINENEWLINE\(\bullet\) If \(\alpha \neq -1\), then the space of \(\mu_{1,\alpha}\)-integrable functions is equal to the space of continuous functions \(\mathbb{Z}_p \rightarrow K\).NEWLINENEWLINE\(\bullet\) The measure \(\mu_{1,-1}\) is equal to the opposite of the Dirac measure at \(0\) and all functions \(\mathbb{Z}_p \rightarrow K\) are \(\mu_{1,-1}\)-integrable.NEWLINENEWLINE\(\bullet\) Suppose that \(p=5\) and let \(\mu(5):= \sum_{\alpha^{4}=1} \, \mu_{1,\alpha}\). Then the space of \(\mu(5)\)-integrable functions is equal to the space of continuous functions \(\mathbb{Z}_5 \rightarrow K\).