Bounded charges and their ranges (Q1101842)

A bounded charge \(\mu\) on a \(\sigma\)-field \({\mathcal A}\) is said to have the strongly closed property if for every A in \({\mathcal A}\) the set \(\{\mu (B): B\in {\mathcal A}\cap {\mathcal P}(A)\}\) is closed. It is proved that this property is characteristic for the bounded continuous charges and for the bounded strongly atomic charges on \(\sigma\)-fields. Moreover the range \(\mu ({\mathcal A}\cap {\mathcal P}(A))\) for every A in \({\mathcal A}\), in this case, is like the range of a bounded measure.