arXiv subjects
Publications and source records attributed to E. Chetcuti.
It is shown that the bona fide generalization of the Vitali-Hahn-Saks Theorem to von Neumann algebras is possible if, and only if, the algebra is finite. This settles the problem on the noncommutative Vitali-Hahn-Saks Theorem completely and provides new means of characterizing finite von Neumann algebras.
Suppose that $S$ is an incomplete inner product space. A. Dvurečenskij shows that there are no finitely additive states on orthogonally closed subspaces, $F(S)$, of $S$ that are regular with respect to finitely dimensional spaces. In this note we show that the most important special case of the former result--the case of the evaluations given by vectors in the ``Gleason manner''--allows for a relatively simple proof. This result further reinforces the conjecture that there are no finitely additive states on $F(S)$ at all.