Pan-Integrals Based on Optimal Measures

The pan-integrals are based on a special type of commutative isotonic semiring ((R) over bar+, circle plus, circle times) and the monotone measures mu defined on a measurable space (X, A). On the other hand, based on a pan-addition circle plus each monotone measure mu generates a new monotone measure mu(circle plus) which is called the circle plus-optimal measure (to mu and circle plus). Such monotone measure mu(circle plus) is greater than or equal to mu and it is super-circle plus-additive (i. e., mu(circle plus)(A boolean OR B) >= mu(circle plus)(A) circle plus mu(circle plus)(B) whenever A, B is an element of A, A boolean AND B = (sic). In this note, we shall present some new properties of the pan-integral. It is shown that the pan-integral with respect to mu coincides with the pan-integral with respect to mu(circle plus) on a given pan-space (X, A, mu, (R) over bar+, circle plus, circle times). As a special case of this result, we show that the.-optimal measure derived from mu is totally balanced for the pan-integrals.