On a class of ultrametric measures (Q2758100)

Let \(C\) be a compact set in an ultrametric space \((X,d)\). In the paper under review, the author associates to \(C\) a real valued and a \(p\)-adic measure, \(\mu _C\). Using this measure the definition of the trace function given by \textit{V. Alexandru, N. Popescu} and \textit{A. Zaharescu} [J. Number Theory 88, 13-38, (2001; Zbl 0965.11049)] is extended. NEWLINENEWLINENEWLINELet \(D[x,r]=\{y\in C\mid d(x,y)\leq r\}\). Let \(\varepsilon _1\) be the diameter of \(C\). For any \(0< \varepsilon< \varepsilon _1\), \(C\) can be covered with at least two balls. Choose \(0 < \varepsilon _2 <\varepsilon _1\) largest such that \(C\) can be covered with the smallest number of balls \(n_2>1\), and so on. We denote by \(S _i\) the finite set of distinct balls \(D[x_j^{(i)},\varepsilon _i]\), \(j=1,\ldots, n_i\) which cover \(C\). The sets \(S_i\) are used to define a real valued measure \(\mu _C\). A function \(f: C\to \mathbb{C}\) is said to be integrable on \(C\) if the set of all its Riemann sums \(\sum _{j=1} ^{n_i} f(x_j^{(i)}) \mu _C(D[x_j^{(i)},\varepsilon _i])\) has a unique limit point in \(\mathbb{C}\). Any integrable function in bounded on \(C\). Let \(\omega (f; D[x_j^{(i)},\varepsilon _i])=\sup \{|f(x)-f(y)|\mid x, y \in D[x_j^{(i)},\varepsilon _i]\}\) and \(\omega(f;i) = \max \{w(f; D[x_j^{(i)},\varepsilon _i])\mid 1\leq j\leq n_i\}\). The author shows that if \(C\) is an infinite compact set and \(f: C \to \mathbb{C}\) is a continuous function such that \(\sum _{i=1}^\infty \omega(f;i) <\infty\), then \(f\) is integrable on \(C\). If the measure \(\mu _C\) has values in the \(p\)-adic complex field \(\mathbb{C}_p\), the integrability is defined similarly, and also it is proved that if \(f\) is continuous on \(C\) with \(p\)-adic values, then \(f\) is \(p\)-adic integrable on \(C\). Let \(T\in \mathbb{C}_p\) be transcendental over \(\mathbb{Q} _p\) and let \(C(T)\) be the orbit of \(T\) with respect to the Galois group \(\text{Gal}(\overline{\mathbb{Q}}_p /\mathbb{Q}_p)\). The results of this paper are applied to \(C(T)\) giving the extended definition of the trace function of Alexandru, Popescu and Zaharescu.