Legend lecture: Current perspectives on choquet calculus

By Calculus (the study of change), we mean Integral and Differential Calculus originated with Newton and Leibniz in 17C. In this talk, we discuss Choquet Calculus which is nonlinear in general. So far most studies on Choquet integrals have been devoted to the discrete case. In Choquet Calculus we deal with continuous Choquet integrals and also derivatives. First we show how to calculate continuous Choquet integrals. To this aim, we consider distorted Lebesgue measures (a class of fuzzy measures), and non-negative and non-decreasing functions; distorted Lebesgue measures are obtained by the monotone transformation of Lebesgue measures. Next we define derivatives of functions with respect to distorted Lebesgue measures. We also discuss the identification of distorted Lebesgue measures. Then we show a relation between Choquet Calculus and Fractional Calculus. Further, we consider differential equations with respect to distorted Lebesgue measures and give their solutions. Lastly we present the concept of conditional distorted Lebesgue measures.