The main result of the paper consists in showing the following theorem being an analogue of the classical result for weights on von Neumann algebras.
Theorem. Let \(\phi\) be a weight on a JBW-algebra A. The following conditions are equivalent:
(i) \(\phi\) is completely additive;
(ii) \(\phi\) is normal;
(iii) \(\phi\) is weakly continuous from below;
(iv) \(\phi (a)=\sup \{\psi (a):\psi \in F\}\), \(a\in A^+\), where F is a family of positive normal functionals on A;
(v) \(\phi\) is a sum of positive normal functionals on A.