On the Lebesgue measurability of continuous functions in constructive analysis.

*(English)*Zbl 0742.03024The 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 The proof depends on a theorem first proved by Demuth in his unpublished thesis in 1964 and later rediscovered by Bridges.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.

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.

Reviewer: H.Schwichtenberg (München)

##### 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 |

##### Keywords:

intuitionism; \(\alpha\)-singular covering; constructive mathematics; Bishop; Russian school; Brouwer; constructive measure theory
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\textit{D. Bridges} and \textit{O. Demuth}, Bull. Am. Math. Soc., New Ser. 24, No. 2, 259--276 (1991; Zbl 0742.03024)

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##### References:

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