On Minimal and Maximal p-operator Space Structures

We show that L∞(µ), in its capacity as multiplication operators on Lp(µ), is minimal as a p-operator space for a decomposable measure μ. We conclude that L1(μ) has a certain maximal type p-operator space structure that facilitates computations with L1(μ) and the projective tensor product.

On Properties of Pseudointegrals Based on Pseudoaddition Decomposable Measures

We mainly discussed pseudointegrals based on a pseudoaddition decomposable measure. Particularly, we give the definition of the pseudointegral for a measurable function based on a strict pseudoaddition decomposable measure by generalizing the definition of the pseudointegral of a bounded measurable function. Furthermore, we got several important properties of the pseudointegral of a measurable function based on a strict pseudoaddition decomposable measure.

On Decomposable Measures Induced by Metrics

We prove that for a given normalized compact metric space it can induce aσ-max-superdecomposable measure, by constructing a Hausdorff pseudometric on its power set. We also prove that the restriction of this set function to the algebra of all measurable sets is aσ-max-decomposable measure. Finally we conclude this paper with two open problems.

PROBABILISTIC MULTI-VALUED CONTRACTIONS AND DECOMPOSABLE MEASURES

A general fixed point theorem for a multi-valued probabilistic q-contraction f : S → 2S is proved, where (S, ℱ, T) is a complete Menger space and ℱ satisfies a growth condition which is connected with the countable extension of a t-norm T. As a corollary a generalization of Tardiff's result27 is obtained. A random fixed point result is proved, where a measure space related to decomposable measure is used.

Extension on the fuzzy integral based on⊕-decomposable measure

We extend the concept of fuzzy integral based on⊕-decomposable measure from nonnegative fuzzy measurable function to extended real-valued fuzzy measurable function. Further investigations of fuzzy integrals based on pseudo-additive decomposable measure are carried out. Meanwhile, the space(S(μ),σ(⋅,⋅))of all fuzzy measurable function will be proved to be a pseudo-metric space. Finally, as an application of this extension the Pettis integral will be obtained for that kind of fuzzy integral.

INTEGRATION WITH RESPECT TO DECOMPOSABLE MEASURES, BASED ON A CONDITIONALLY DISTRIBUTIVE SEMIRING ON THE UNIT INTERVAL

To cover almost all known Lebesgue type integrals constructed by means of pseudo-operations, the (S, U)-integral based on a t-conorm S, a uninorm (or t-norm) U and an S-measure (decomposable measure) is introduced and its properties are studied. Also its relationship to aggregation operators is discussed.

PSEUDO-CONVOLUTION BASED ON IDEMPOTENT OPERATION AS LIMIT OF g-CONVOLUTION

Operation with functions known as pseudo-convolution and its generalization as well as theirs basic properties has been presented. Then, it has been proved that pseudo-convolution which core is pseudo-integral based on max or min decomposable measure can be obtained as limit of g-convolutions, i.e., pseudo-convolutions with pseudo-integrals based on ⊕-decomposable measures where ⊕ is generated pseudo-addition, as their cores.