Infinite combinatorics and the foundations of regular variation |
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Authors: | NH Bingham AJ Ostaszewski |
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Institution: | aMathematics Department, Imperial College London, London SW7 2AZ, United Kingdom;bMathematics Department, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom |
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Abstract: | The theory of regular variation is largely complete in one dimension, but is developed under regularity or smoothness assumptions. For functions of a real variable, Lebesgue measurability suffices, and so does having the property of Baire. We find here that the preceding two properties have common combinatorial generalizations, exemplified by ‘containment up to translation of subsequences’. All of our combinatorial regularity properties are equivalent to the uniform convergence property. |
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Keywords: | Regular variation Uniform convergence theorem Cauchy functional equation Baire property Measurability Density topology Measure-category duality Infinite combinatorics Subuniversal set No Trumps principle |
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