The Begümian ς-Structure: A Unified Framework of Sets, Functions, and Probability

This paper introduces the ς-structure, a novel mathematical object that unifies the concepts of sets, functions, and probability into a single formalism. The ς-structure, denoted by the Greek final sigma (ς), is defined as an ordered triple consisting of a set, a function, and a probability distribution. The main operation, called the ς-sum, generalizes the expected value by weighting function values over a set according to a given probability distribution. We provide rigorous definitions, key theorems with proofs, illustrative examples, and a discussion of possible generalizations and applications across mathematics and science. The new symbol ς is proposed as the canonical notation for this hybrid structure.

Originality and AI-use statement:
This work is an original research output by Begüm Yıldırım. AI tools, if used, were limited to language refinement, grammar correction, formatting, translation assistance, and clarity improvement. The conceptual framework, research direction, interpretation, models, and conclusions belong to the author. External sources, datasets, or prior works are cited where applicable.