State-complete Riesz MV-algebras are a particular class of probability MV-algebras. We associate to any state-complete Riesz MV-algebra (A,s) a measure space (X,Ω,μ) such that (A,s) and (L 1(μ) u,s μ ) are isometrically isomorphic Riesz MV-algebras, where L 1(μ) u is an interval of L 1(μ) and s μ is the integral. This result can be seen as an analogue of Kakutani's concrete representation for
... [Show full abstract] L-spaces [10] and it leads to a categorical duality between Riesz MV-algebras and a special class of measure spaces (called L-measure spaces).