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From Sets and Types to Topology and AnalysisTowards practicable foundations for constructive mathematics$
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Laura Crosilla and Peter Schuster

Print publication date: 2005

Print ISBN-13: 9780198566519

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198566519.001.0001

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APPROXIMATING INTEGRABLE SETS BY COMPACTS CONSTRUCTIVELY

APPROXIMATING INTEGRABLE SETS BY COMPACTS CONSTRUCTIVELY

Chapter:
(p.268) 17 APPROXIMATING INTEGRABLE SETS BY COMPACTS CONSTRUCTIVELY
Source:
From Sets and Types to Topology and Analysis
Author(s):

Bas Spitters

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566519.003.0017

The interplay between intuitionistic mathematics and Bishop-style constructive mathematics is crucial for this chapter's contribution. In locally compact spaces, (Borel-)measurable sets can be approximated by compact sets. Ulam extended this result to complete separable metric spaces. This chapter gives a constructive proof of Ulam's theorem. The technique used aims to first prove the theorem intuitionistically and then, using the logical ‘trick’ seen in Chapter 16, to obtain a proof which is acceptable in Bishop-style mathematics. The proof also provides some insight into the trick seen in Chapter 16. Finally, it shows how several intuitionistic measure theoretic theorems can be extended to regular integration spaces, that is, integration spaces where integrable sets can be approximated by compacts. These results may also help in understanding Bishop's original choice of definitions.

Keywords:   Bishop-style constructive mathematics, intuitionistic mathematics, Borel measurability, Ulam theorem, metric spaces

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