A Generalized Mathematical Framework Based on Measurable Spaces

This paper presents a generalized mathematical framework based on measurable spaces for a unified mathematical description of engineering processes characterized by measurable quantitative properties. The proposed approach is founded on the measurable structure (A, Σ, μ, X), where measurable objects, a quantitative measure, and the corresponding mapping form the basis for describing heterogeneous engineering systems within a common mathematical setting. The mathematical consistency of the proposed framework is established through rigorous theoretical analysis. In particular, the principal analytical properties of the cumulative measure generated by the measurable structure are derived. The obtained results establish its monotonicity and identify conditions under which it remains constant, thereby providing a theoretical justification for employing the proposed framework as a reliable basis for mathematical modelling and analytical investigation. To illustrate the applicability of the developed theory, a representative mechanical example is considered. The example demonstrates that the proposed framework naturally describes a practical engineering process without requiring any modification of the underlying mathematical structure, thereby confirming its practical interpretability while preserving its general character. The principal contribution of the study consists in the development of a unified mathematical framework that separates the abstract measurable structure from its physical interpretation. Such an approach provides a common theoretical foundation for the mathematical description of engineering systems and creates opportunities for further development of specialized models and computational methods in various areas of mechanics and engineering.