Automatic boundedness of quantum measures (Q1851399)

Let \(G\) be a a Hausdorff topological commutative group. The authors call a subset \(B\) of \(G\) strictly bounded if for every \(0\)-neighbourhood \(U\) in \(G\) there is a natural number \(n\) such that \(B\subset \{\sum_{i=1}^n x_i: x_i\in U\}\) (i.e. \(B\) is \`\` bounded\'\'\quad in the sense of \textit{D. Landers} and \textit{L. Rogge} [Manuscr. Math. 4, 351-359 (1971; Zbl 0217.14902)]) Let \(L\) be a complete orthomodular lattice and \(\mu:L\rightarrow G\) a completely additive measure. It is proved that \(\mu(L)\) is stricly bounded iff for every atom \(e\) of the centre \(C(L)\) of \(L\) \(\mu([0,e])\) is strictly bounded. In particular \(\mu(L)\) is bounded if \(C(L)\) is atomless.