APPROXIMATING INTEGRABLE SETS BY COMPACTS CONSTRUCTIVELY
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.
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