We study a Fuzzy type integral for measurable multifunctions with respect to a Fuzzy measure. Some classical properties and convergence theorems are presented.

In this paper, we introduce Fuzzy measure and Fuzzy integral concepts and express some of the Fuzzy integral properties. The main purpose of this article is to reviewing of  some important mathematical inequalities that have many applications in modeling mathematical problems. Firstly, we prove  the related Gauss-Winkler type inequality for Fuzzy integrals. Indeed, we prove Fuzzy version provided by D. H. Hong. Another the famous mathematical inequality is Minkowski's inequality. It is an important inequality from both mathematical and application points of view. Here, we state  a Minkowski type inequality for Fuzzy integrals. The established results are based on the classical Minkowski's inequality for integrals. In the continue, we showed that by an example the classical Pr\'{e}kopa-Leindler type inequality is not valid for the Sugeno integral. We proved one version of the Pr\'{e}kopa-Leindler type inequality by adding  concave Fuzzy measure and quasi-concave Fuzzy measure assumptions for the Sugeno integral  with different proofs. Also, we  obtained a derivation version of the Pr\'{e}kopa-Leindler inequality and illustrated all of the main results by examples. Finally, we investigate the Thunsdorff's inequality for Sugeno integral. By an example, we show that the classical form of this inequality does not hold for the Sugeno integral. Then, by reviewing the initial conditions, we prove two main theorems for this inequality.By checking the special case of the aforementioned Thunsdorff's inequality, we prove Frank-Pick type inequality for the Sugeno integral and illustrate it by an example.

The Fuzzy integrals are a kind of Fuzzy measures acting on Fuzzy sets. They can be viewed as an average membership value of Fuzzy sets. The value of the Fuzzy integral in a decision making environment where uncertainty is present has been well established. Most of the integral inequalities studied in the Fuzzy integration context normally consider conditions such as monotonicity or comonotonicity. In this paper, we are trying to extend the Fuzzy integrals to the concept of concavity. It is shown that the Hermite-Hadamard integral inequality for concave functions is not satisfied in the case of Fuzzy integrals. We propose upper and lower bounds on the Fuzzy integral of concave functions. We present a geometric interpretation and some examples in the framework of the Lebesgue measure to illustrate the results.