Theory of capacities

Theory of capacities

1954 | Gustave Choquet
Gustave Choquet's work on the theory of capacities, published in the *Annales de l’institut Fourier* in 1954, addresses the problem of whether the interior Newtonian capacity of an arbitrary Borel subset \(X\) of \(\mathbb{R}^3\) is equal to the exterior Newtonian capacity of \(X\). He systematically studies non-additive set functions and identifies particularly interesting classes, aiming to establish a theory analogous to classical measure theory. Choquet shows that the classical Newtonian capacity \(f\) belongs to one of these classes, specifically proving that for any compact subsets \(A\) and \(B\) of \(\mathbb{R}^3\): \[ f(A \cup B) + f(A \cap B) \leq f(A) + f(B). \] This result implies that every Borel and analytic set is capacitable with respect to the Newtonian capacity. The work also explores the properties of alternating and monotone functions, their integral representations, and the extension and restriction of capacities. In the first chapter, Choquet redefines Borel and analytic sets in topological spaces, emphasizing the use of compact sets and continuous mappings to avoid irregular topological characteristics. He introduces the concept of K-borelian and K-analytic sets, which are generated from compact sets using countable intersections and unions. The second chapter defines Newtonian and Greenian capacities for compact sets and arbitrary sets, associating equilibrium potentials and capacities with compact subsets. It proves that the sequence of inequalities involving these potentials is complete, and discusses the differential of \(f\) with respect to suitable increments. The third chapter introduces several classes of functions, including alternating and monotone functions, and their properties. It explores the extension and restriction of capacities, showing that these operations preserve certain classes of capacities. The fourth chapter delves into the extremal elements of convex cones and integral representations, using the Krein-Milman theorem to establish the existence of integral representations for certain functions. The final chapter applies these results to measure theory and monotone capacities, proving that all Borel subsets of a complete metric space and complements of analytic sets are capacitable with respect to monotone capacities. Overall, Choquet's work provides a comprehensive framework for understanding and extending the theory of capacities, with applications to various mathematical fields.Gustave Choquet's work on the theory of capacities, published in the *Annales de l’institut Fourier* in 1954, addresses the problem of whether the interior Newtonian capacity of an arbitrary Borel subset \(X\) of \(\mathbb{R}^3\) is equal to the exterior Newtonian capacity of \(X\). He systematically studies non-additive set functions and identifies particularly interesting classes, aiming to establish a theory analogous to classical measure theory. Choquet shows that the classical Newtonian capacity \(f\) belongs to one of these classes, specifically proving that for any compact subsets \(A\) and \(B\) of \(\mathbb{R}^3\): \[ f(A \cup B) + f(A \cap B) \leq f(A) + f(B). \] This result implies that every Borel and analytic set is capacitable with respect to the Newtonian capacity. The work also explores the properties of alternating and monotone functions, their integral representations, and the extension and restriction of capacities. In the first chapter, Choquet redefines Borel and analytic sets in topological spaces, emphasizing the use of compact sets and continuous mappings to avoid irregular topological characteristics. He introduces the concept of K-borelian and K-analytic sets, which are generated from compact sets using countable intersections and unions. The second chapter defines Newtonian and Greenian capacities for compact sets and arbitrary sets, associating equilibrium potentials and capacities with compact subsets. It proves that the sequence of inequalities involving these potentials is complete, and discusses the differential of \(f\) with respect to suitable increments. The third chapter introduces several classes of functions, including alternating and monotone functions, and their properties. It explores the extension and restriction of capacities, showing that these operations preserve certain classes of capacities. The fourth chapter delves into the extremal elements of convex cones and integral representations, using the Krein-Milman theorem to establish the existence of integral representations for certain functions. The final chapter applies these results to measure theory and monotone capacities, proving that all Borel subsets of a complete metric space and complements of analytic sets are capacitable with respect to monotone capacities. Overall, Choquet's work provides a comprehensive framework for understanding and extending the theory of capacities, with applications to various mathematical fields.
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