Every state on semisimple MV-algebra is integral

Integral representation theorem will be established for finitely additive probability measures (states) on semisimple MV-algebras. This result generalizes the well-known theorem of Butnariu and Klement in case of σ-order continuous states on tribes of fuzzy sets. Precisely, it will be demonstrated that every state on a separating clan of continuous fuzzy sets arises as an integral with respect to a unique Borel probability measure. The key technique leading to this result exploits the geometrical–topological properties of the state space: the set of all states on every MV-algebra forms a Bauer simplex.