Three problems of Aronszajn in measure theory (Q1065184)

In the paper the solution of three problems appearing in: \textit{N. Aronszajn}, Stud. Math. 57, 147-190 (1976; Zbl 0342.46034), is given. One of the problems is: to describe all the measures which are absolutely continuous with respect to \({\mathcal U}\{a_ n\}\) for a given sequence \(\{a_ n\}.\) We clarify the necessary notions: \(X\) denotes a separable Banach space, \(B(X)\) the \(\sigma\)-algebra of Borel sets of \(X\). For a given sequence \(\{a_ n\}\) of elements of X the set \({\mathcal U}\{a_ n\}\) denotes the set of such \(B\in B(X)\) for which \(B=\cup_{n}B_ n\), where \(B_ n\in B(X)\) and the usual Lebesgue measure of \((B_ n+x)\cap R^ la_ n\) is equal to zero. Here \(R^ la_ n\) is the line determined by \(a_ n\). The absolute continuity of a measure \(\mu\) with respect to \({\mathcal U}\{a_ n\}\) means that \(\mu\) vanishes on every \(A\in {\mathcal U}\{a_ n\}\).