Measure and integration theory uses the axiom of choice extensively, for example the idea behind σ-algebra is that there are non-measurable sets (in the sense of lebesgue) like Vitali set, but on the other hand Solovay has prove that their exist model of ZF without axiom of choice in which every set of reals is measurable.
This is why I ask myself if you know if there are courses in measure theory and integration that do not use the axiom of choice or even better that are based on intuitionism ?
Sorry if my question is not very clear, I am not an expert in intuitionism, just a bit curious about it.