On the setwise convergence of sequences of signed topological measures (Q1942315)

The author continues the study of proper quasi-measures and signed quasi-measures started in two earlier papers [Math. Notes 81, No. 5, 671--680 (2007); translation from Mat. Zametki 81, No. 5, 751--759 (2007; Zbl 1210.28003); ``A signed quasi-measure decomposition'', Vestn., Samar.,Gos. Univ. Estestvennonauchen. Ser. 62, 192--207 (2008)]. Quasi-measures are also called topological measures. A topological measure is an \(R^{+}\)-valued set function on the class \(\alpha\) of open subsets (\(\tau\)) and closed subsets (\(\zeta\)) of a compact Hausdorff space \(X\), which is finitely additive and regular; a proper topological measure is a topological measure \(\mu\) such that if \(\nu\) is a regular Borel measure on \(X\) with \(0 \leq \nu \leq \mu \) then \(\nu =0\). These definitions are extended to signed topological measures and proper signed topological measures. \(STM\) and \(PSTM\) denote the classes of signed topological measures and proper signed topological measures; \(SM\) is the class of signed regular Borel measures. The proved main results are the following. I. Let \(\{ \lambda_{n} \} \subset SM\) and \(\{ \nu_{n} \} \subset PSTM\) be two sequences such that \(\lim (\lambda_{n} + \nu_{n})(E) =0\;\forall E \in \alpha\). Then \(\lim \lambda_{n}(E) =0\) and \(\lim \nu_{n}(E) =0\;\forall E \in \alpha\). II. Suppose that \(\{ \mu_{n} \} \subset PSTM\) is a uniformly bounded sequence and converges to \(\mu\) pointwise on \(\alpha\), then \( \mu \in PSTM\). The result remains valid if \( PSTM\) is replaced by \( PTM\). III. Let \(\{ \lambda_{n} \} \subset SM\), \(\{ \mu_{n} \} \subset STM\) and \(\{ \nu_{n} \} \subset PSTM\) be three sequences such that \( \mu_{n} = \lambda_{n}+ \nu_{n} \). Assume \(\lim \lambda_{n}(E) = \lambda(E)\), \( \lim \mu_{n}(E) = \mu(E)\) and \(\lim \nu_{n}(E) = \nu(E)\;\forall E \in \alpha\). Then \( \lambda \in SM\) and \(\mu\), \(\nu\) are finitely additive and regular.