Annals of Pure and Applied Logic 30 (2):173-200 (1986)

Abstract
A theory of approximation to measurable sets and measurable functions based on the concepts of recursion theory and discrete complexity theory is developed. The approximation method uses a model of oracle Turing machines, and so the computational complexity may be defined in a natural way. This complexity measure may be viewed as a formulation of the average-case complexity of real functions—in contrast to the more restrictive worst-case complexity. The relationship between these two complexity measures is further studied and compared with the notion of the distribution-free probabilistic computation. The computational complexity of the Lebesgue integral of polynomial-time approximable functions is studied and related to the question “FP = ♯P?”
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DOI 10.1016/0168-0072(86)90005-9
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References found in this work BETA

Nicht Konstruktiv Beweisbare Sätze der Analysis.Ernst Specker - 1949 - Journal of Symbolic Logic 14 (3):145-158.
On the Definition of Computable Function of a Real Variable.J. C. Shepherdson - 1976 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 22 (1):391-402.
On the Definition of Computable Function of a Real Variable.J. C. Shepherdson - 1976 - Mathematical Logic Quarterly 22 (1):391-402.

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