Fuzzy Parameterized Fuzzy Soft Measure Spaces

This study first defines some families of fuzzy parameterized fuzzy soft sets (fpfs-sets) such as fpfs-rings, fpfs-$\\sigma$-rings, fpfs-algebras, fpfs-$\\sigma$-algebras, fpfs-$\\pi$-systems, and fpfs-$\\lambda$-systems. It also investigates some fundamental properties of these fpfs-set families and analyzes their relations with each other. Then, this study proposes the concept of fpfs-pre-measures based on fpfs-$\\sigma$-rings and exemplifies them. Subsequently, it defines fpfs-measures based on fpfs-$\\sigma$-algebras, illustrates them by examples, and studies some of their basic properties. Moreover, this article propounds several types of functions, defined from an fpfs-$\\sigma$-algebra to the extended real numbers and referred to as additive, finitely additive, null additive, subadditive, subtractive, and so on, and investigates some of their basic properties. Afterward, it suggests fpfs-outer measures, exemplifies them, and explores some of their essential properties. Furthermore, this article offers $m^*$-fpfs-measability and analyzes some of its key properties. Finally, it addresses whether further research into these aspects is warranted.