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@article{175817,
        author = {Dr. S.K. Pandey and Kapil Kumar Singh and Harshraj Shukla},
        title = {Hausdorff Measurable Multifunctions, Theory, Properties, and their Applications},
        journal = {International Journal of Innovative Research in Technology},
        year = {2025},
        volume = {11},
        number = {11},
        pages = {7969-7981},
        issn = {2349-6002},
        url = {https://ijirt.org/article?manuscript=175817},
        abstract = {This paper comprehensively studies Hausdorff's measurable multifunctions, a concept at the intersection of measure theory, topology, and set-valued analysis. The Hausdorff measure extends classical notions of dimension and size to sets with complex or fractal structures. Multifunctions, or set-valued functions, map each input to a set rather than a single value and arise naturally in areas like optimization, control theory, and economics.
The paper defines Hausdorff measurability in terms of the Hausdorff metric, focusing on the measurability of excess functions that quantify the "distance" between sets. It explores relationships between Hausdorff, weak, strong, and Effros measurability, highlighting their equivalence under certain conditions, especially in separable metric spaces.
Key properties such as continuity-like behaviors, closure under operations, and convergence of sequences are analyzed. The existence of measurable selections—single-valued functions chosen from multifunctions—is emphasized due to its importance in applications.
Practical applications span optimization problems with uncertainty, differential inclusions in control systems, game theory, fractals, and stochastic processes. The paper also references foundational theorems and literature that provide tools for studying and applying Hausdorff measurable multifunctions.},
        keywords = {Hausdorff measurability, Hausdorff metric, stochastic processes, Hausdorff's measurable multifunction,},
        month = {May},
        }