The Haagerup problem on subadditive weights on $W^*$-algebras

In year 1975 U. Haagerup posed the following question: Whether every normal subadditive weight on a $W^*$-algebra is sigma-weakly lower semicontinuous? Here we positively answer this question in a particular case of abelian $W^*$-algebras and present a general form of normal subadditive weights on these algebras.