# zbMATH — the first resource for mathematics

On the Lebesgue measurability of continuous functions in constructive analysis. (English) Zbl 0742.03024
The paper starts with a general discussion of three main approaches to constructive mathematics: that of Bishop (BISH), that of the Russian school (RUSS) and the historically first, i.e. that of Brouwer (INT). The intended reader is a mathematician with little or no knowledge of the technicalities of constructive mathematics. It is argued that RUSS as well as INT can be viewed as BISH with certain adjoined principles. This view makes it possible to prove independence results relative to BISH. The main results of the paper give examples of propositions independent of BISH, e.g.: There is a bounded, pointwise continuous map of [0,1] into $$\mathbb{R}$$ that is not Lebesgue measurable. The proof depends on a theorem first proved by Demuth in his unpublished thesis in 1964 and later rediscovered by Bridges.
Apart from proving this interesting result the paper gives a good introduction into constructive measure theory, a central part of modern constructive mathematics. It provides detailed references to the literature.

##### MSC:
 03F60 Constructive and recursive analysis 03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations 03F55 Intuitionistic mathematics 03F65 Other constructive mathematics 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 03F25 Relative consistency and interpretations
Full Text:
##### References:
 [1] Michael J. Beeson, Foundations of constructive mathematics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 6, Springer-Verlag, Berlin, 1985. Metamathematical studies. · Zbl 0565.03028 [2] Errett Bishop and Douglas Bridges, Constructive analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 279, Springer-Verlag, Berlin, 1985. · Zbl 0656.03042 [3] Douglas Bridges and Fred Richman, Varieties of constructive mathematics, London Mathematical Society Lecture Note Series, vol. 97, Cambridge University Press, Cambridge, 1987. · Zbl 0618.03032 [4] Osval$$^{\prime}$$d Demut, On Lebesgue integration in constructive analysis, Dokl. Akad. Nauk SSSR 160 (1965), 1239 – 1241 (Russian). [5] Osval$$^{\prime}$$d Demut, On Lebesgue integration in constructive analysis, Dokl. Akad. Nauk SSSR 160 (1965), 1239 – 1241 (Russian). [6] Osval$$^{\prime}$$d Demut, The Lebesgue integral in constructive analysis, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 4 (1967), 30 – 43 (Russian). [7] Osvald Demuth, The Lebesgue integral and the concept of measurability of functions in constructive analysis, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 8 (1968), 21 – 28 (Russian). [8] Osval$$^{\prime}$$d Demut, The differentiability of constructive functions, Comment. Math. Univ. Carolinae 10 (1969), 167 – 175 (Russian). [9] O. Demut, The spaces \?_{\?} and \? in constructive mathematics, Comment. Math. Univ. Carolinae 10 (1969), 261 – 284 (Russian). [10] O. Demuth and A. Kučera, Remarks on constructive mathematical analysis, Logic Colloquium ’78 (Mons, 1978) Stud. Logic Foundations Math., vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 81 – 129. [11] A. Heyting, Intuitionism-an introduction, 3rd ed., North-Holland, Amsterdam, 1971. · Zbl 0219.02013 [12] A. J. Kfoury, Robert N. Moll, and Michael A. Arbib, A programming approach to computability, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1982. AKM Series in Theoretical Computer Science. · Zbl 0497.68025 [13] B. A. Kushner, Lectures on constructive mathematical analysis, Translations of Mathematical Monographs, vol. 60, American Mathematical Society, Providence, RI, 1984. Translated from the Russian by E. Mendelson; Translation edited by Lev J. Leifman. · Zbl 0547.03040 [14] Hartley Rogers Jr., Theory of recursive functions and effective computability, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.