The Radon-Nikodým theorem for fuzzy probability spaces (Q1185030)

The Radon-Nikodým theorem is proved for fuzzy observables. Some possibilities of applications are given. In my opinion, the signed measure is not a real generalization of fuzzy \(P\)-measure because of: at the first --- the case \(m(\mathbf{1}_ x)=0\) is not interested; at the second --- for each signed measure \(m\) such that \(m(\mathbf{1}_ x)<0\) we have signed measure \(m'=-m\); and at the third --- if \(m(\mathbf{1}_ x)>0\), then the considered signed measure \(m\) we may equivalently replace by the signed measure \(m'(A)=m(A)/m(\mathbf{1}_ x)\) which is a fuzzy \(P\)-measure.