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Annali di Matematica Pura ed Applicata

, Volume 118, Issue 1, pp 217–227 | Cite as

Integrability and closest approximation representations

  • William D. L. Appling
Article

Summary

Suppose U is a set,F is a field of subsets of U, andp AB is the set of all real-valued bounded finitely additive functions defined onF. This paper consists of two main parts. In the first, a previously given (seeRiv. Math. Univ. Parma, (3),2(1973), pp. 251–276) notion of a subset ofp AB defined by certain closure properties and called a C-set, is considered, and those C-sets that are linear spaces are characterized. Now, suppose γ is a function whose domain isF and whose range is a collection of number sets with bounded union. The set,Jγ, of all elements ofp AB with respect to which γ is integrable, for refinements of subdivisions, is a C-set and a linear space (seeRend. Sem. Mat. Univ. Padova,52 (1974), pp. 1–24). The second part of this paper concerns, for μ inp AB and nonnegative-valued, a representation of the element ofJγ closest to μ with respect to variation norm.

Keywords

Main Part Linear Space Variation Norm Additive Function Approximation Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [1]
    W. D. L. Appling,A generalization of absolute continuity and of an analogue of the Lebesgue decomposition theorem, Riv. Mat. Univ. Parma, (3),2 (1973), pp. 251–276.zbMATHMathSciNetGoogle Scholar
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    W. D. L. Appling,Upper and lower integral difference functionals, closest approximations, and integrability, Rend. Sem. Mat. Univ. Padova,52 (1974), pp. 1–24.MathSciNetGoogle Scholar
  3. [3]
    E. Hellinger,Die Orthogonalinvarianten quadratischer Formen von unendlichvielen Variablen, Diss. Gottingen (1907).Google Scholar
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    A. Kolmogoroff,Untersuchungen ober den Integralbegriff, Math. Ann.,103 (1930), pp. 654–696.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1978

Authors and Affiliations

  • William D. L. Appling
    • 1
  1. 1.DentonU.S.A.

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