Markov type inequalities for fuzzy integrals
Given a fuzzy measure space \((X,\Sigma,\mu)\) and a \(\mu\)-measurable nonnegative function \(f:X\to[0,\infty)\), a typical result from the paper is the following inequality: For \(0<c\leq1\), \[ \mu\{x\in A:f(x)\geq c\}\leq\frac{1}{c} \fint_A f(t)dt, \] where \(-\!\!\!\!\!\int\) stands dor the Sugeno fuzzy integral defined by \[ \fint_A f(t)dt := \sup_{\alpha\geq 0}\min\big(\alpha,\mu\{x\in A:f(x)\geq\alpha\}\big). \] This inequality can be regarded as the analogue of Markov's inequality for the Sugeno integral setting. The paper also contains various generalizations of the above inequality and fundamental properties of the Sugeno integral are established.
scientific article; zbMATH DE number 5522209
scientific article