The distribution functions of finitely additive vector measures over \({\mathbb{R}}^ q\). I (Q800517)

Let X be a Banach space and \({\mathcal P}={\mathcal P}^ q\) be the pre-ring of bounded half-open half-closed sub-intervals of \(\Omega =R^ q\), \(q\geq 1\). The author considers finitely additive measures on \({\mathcal P}^ q\) with values in X. The following definition of ultra-regularity is introduced: (Def. 2.2..) A finitely additive measure \(\rho\) on a pre-ring \({\mathcal P}\) is ultra-regular iff for each \(P\in {\mathcal P}^ q\) and each \(\epsilon >0\) there exists a compact set K and an open set U in \(\Omega\) such that \(K\subseteq int P\subseteq\bar P\subseteq U\) and for any \(Q\in {\mathcal P}\) such that \(K\subseteq Q\subseteq U\) the estimation \(|\rho(Q)-\rho(P)| <\epsilon\) holds. For a measure on \({\mathcal P}\) the distribution function is defined by \(x(t)=(sgn t)\rho(t^-,t^+]\) where \(t^{\pm}\) are ''positive'' and ''negative'' parts of the vector \(t\in R^ q\), and the interval \((t_ 1,t_ 2]\) is defined as the set: \(\{a\in R^ q| t^ i_ 1<a^ i\leq t^ i_ 2,\quad i=1,2,...q\}.\) For the ultra-regular measures on \(R^ q\) the following main theorem is proved (Thm. 2.7): The conditions: \(\alpha)x(.)\) is continuous on \(R^ q\), \(\beta)\) For any \((a,b]\in {\mathcal P}^ q \lim_{s\to a,t\to b}\rho(t,s]=\rho(a,b]\), \(\gamma)\) \(\rho(.)\) is ultra- regular on \({\mathcal P}^ q\), are equivalent.