Approximating integrable sets by compacts constructively

In locally compact spaces, (Borel-)measurable sets can be approximated by compact sets. Ulam extended this result to complete separable metric spaces. We give a constructive proof of Ulam's theorem. It is first proved intuitionistically and then, using a logical 'trick' due to Ishihara, a proof acceptable in Bishop-style mathematics is obtained. We feel this proof provides some insight into Ishihara's trick. Finally, we show 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 help to understand Bishop's choice of definitions.