We present a general treatment of measures and integrals in certain (monoidal closed) categories. Under appropriate conditions the integral can be defined by a universal property, and the universal measure is at the same time a universal multiplicative measure. In the multiplicative case this assignment is right adjoint to the formation of the Boolean algebra of idempotents. Now coproduct preservation yields an approach to product measures.
Keywords: internal Boolean algebra, universal measure, multiplicative measure, product measure, Boolean algebra of idempotents, symmetric monoidal closed category, cartesian closed category
2000 MSC: 06E05 16A32, 18A15, 18A30, 18A35, 18A40, 18E05, 28A30, 28A33, 28A40, 28A45, 46G10
Theory and Applications of Categories,
Vol. 23, 2010,
No. 12, pp 243-250.