INTEGRABLE FUNCTIONS FOR BERNOULLI MEASURES OF RANK 1 — II

Abstract

In [Integrable functions for Bernoulli measures of rank 1, Ann. Math. Blaise Pascal17 (2010) 349–363; Mesures p-adiques et suites classiques de nombres. Thèse de Doctorat, Université de Bamako, (2011)] we have begun the study of non-Archimedean integrable functions with respect to the normalized Bernoulli measures of rank 1, denoted by μ1,α. The integrability is within the theory of non-Archimedean integration due to Monna and Springer [Intégration non-archimédienne. II, Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag. Math.25 (1963) 643–653; Intégration non-archimédienne. I, Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag. Math.25 (1963) 634–642], which we will call the Monna–Springer theory. In this paper which contains strictly the results of [Integrable functions for Bernoulli measures of rank 1, Ann. Math. Blaise Pascal17 (2010) 349–363; Mesures p-adiques et suites classiques de nombres. Thèse de Doctorat, Université de Bamako (2011)], we cover the integrability with respect to all normalized Bernoulli measures of rank 1. We show that, the integrable functions are reduced to the continuous functions when α is a p-adic unit different from 1 and -1, and all p-adic functions are integrable for α ∈ {-1, 1}. Again following the Monna–Springer theory, we also show that for the measure μ(5) = ∑α4=1 μ1,α, the space of integrable functions is equal to the space of continuous functions.