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Set functions, finite additivity and distribution functions

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Summary

SupposeF is a field of subsets of a set U, α is a function fromF into exp (ℝ), and μ is a nonnegative-valued, finitely additive function fromF into ℝ. In this paper a distribution function, gα, of α with respect to μ is defined. The μ-summability of α is characterized in terms of gα. A representation for the « μ-summability operator for h(α) evaluated on U », where h is a continuous function from ℝ into ℝ such that {|h(x)|/|x| : 1⩽|x|} is bounded, is given as the integral

$$\int\limits_{ - \infty }^\infty {h(t)dg_\alpha (t)} .$$

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References

  1. [1]

    W. D. L. Appling,Summability of real-valued set functions, Riv. Mat. Parma, (2),8 (1967), pp. 77–100.

  2. [2]

    W. D. L. Appling,Continuity and set function summability, Ann. di Mat. pura ed appl. (IV),87 (1972), pp. 357–374.

  3. [3]

    A. Kolmogoroff,Untersuchungen uber den Integralbefriff, Mat. Ann.,103 (1930), pp. 654–696.

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Entrata in Redazione il 18 marzo 1972.

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Appling, W.D.L. Set functions, finite additivity and distribution functions. Annali di Matematica 96, 265–287 (1973) doi:10.1007/BF02414845

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Keywords

  • Distribution Function
  • Continuous Function
  • Additive Function
  • Finite Additivity