Pan-Integrals Based on Optimal Measures

Abstract

The pan-integrals are based on a special type of commutative isotonic semiring $(\overline{R}_+, \oplus , \otimes )$ and the monotone measures $\mu$ defined on a measurable space $(X,\mathcal {A})$. On the other hand, based on a pan-addition $\oplus$ each monotone measure $\mu$ generates a new monotone measure $\mu _{\oplus }$ which is called the $\oplus$-optimal measure (to $\mu$ and $\oplus$). Such monotone measure $\mu _{\oplus }$ is greater than or equal to $\mu$ and it is super-$\oplus$-additive (i.e., $\mu _{\oplus }(A\cup B) \ge \mu _{\oplus }(A)\oplus \mu _{\oplus }(B)$ whenever $A,B\in \mathcal {A}$, $A\cap B=\emptyset$). 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 $\mu _{\oplus }$ on a given pan-space $(X,\mathcal {A},\mu ,\overline{R}_+,\oplus ,\otimes )$. As a special case of this result, we show that the $\oplus$-optimal measure derived from $\mu$ is totally balanced for the pan-integrals.