The Ward property for a \(\mathcal P\)-adic basis and the \(\mathcal P\)-adic integral.
Let \({\mathcal P}= \{p_j\}^\infty_{j=0}\) be a fixed sequence of integers, with \(p_j> 1\) for \(j\in\mathbb{N}\cup \{0\}\). We set \(m_0= 1\) and \(m_k= p_0p_1\cdots p_{k-1}\) for \(k\geq 1\). For fixed \(k\in\mathbb{N}\cup \{0\}\), the closed intervals \[ \Biggl[{r\over m_k},\, {r+1\over m_k}\Biggr]= I^{(k)}_r,\qquad r=0,1,\dots, m_k- 1, \] are called the \({\mathcal P}\)-adic intervals. The set of all such intervals is denoted by \({\mathcal I}_{{\mathcal P}}\). For a given positive function \(\delta\) on \([0, 1]\), we set \[ {\mathcal P}_\delta= \{(I, x): I\in{\mathcal I}_{{\mathcal P}},\, x\in I\subset(x- \delta(x), x+\delta(x)),\, x\in E\}. \] The family \(\{{\mathcal P}_\delta\}_\delta\), where \(\delta\) runs over the set of all positive functions on \([0,1]\), is called the \({\mathcal P}\)-adic basis on \([0, 1]\). It is a differentiation basis. In this paper, it is shown that a \({\mathcal P}\)-adic basis possesses the Ward property if and only if the sequence by which it is defined is bounded. Recall that a basis \(\beta\) satisfies the Ward property whenever each additive interval function is \(\beta\)-differentiable almost everywhere on the set of all points at which at least one of its extreme \(\beta\)-derivatives is finite. Properties (for examples, full descriptive characterizations, VBG and the Lusin condition) of Henstock-Kurzweil type integrals with respect to a \({\mathcal P}\)-adic basis are discussed.
scientific article; zbMATH DE number 1992567
scientific article