Picture fuzzy soft σ -algebra and picture fuzzy soft measure and their applications to multi-criteria decision-making

The concept of picture fuzzy soft sets (PFSSs) has been introduced as an efficacious extension of intuitionistic fuzzy soft sets to model uncertain information, especially when dealing with the responses “Yes”, “No”, and “Abstain”. Recently, Memiş [Another View on Picture Fuzzy Soft Sets and Their Product Operations with Soft Decision-Making. J New Theory 38(2022):1–13, 2022] has redefined the PFSSs and their basic operations to overcome some inconsistencies in the concept. The literature employs the aforesaid inconsistencies in some of Cuong’s definitions. Benefiting from these two inspiring studies, we herein first aim to introduce the picture fuzzy soft σ -algebra, picture fuzzy soft measure, and picture fuzzy soft exterior measure. We then present the famous “Dynkin π – λ theorem” under the picture fuzzy soft environment using the notions of π - and λ -systems. Afterwards, we provide a number of associated results accompanied by explanatory examples. Thereafter, we propose a multi-criteria decision-making method utilising the max-min-max composition on picture fuzzy soft mapping and show how to apply the method to a decision-making scenario about determining the best student semester-wise. Finally, we discuss the need for further research.