There are apparently some difficulties generalising the Henstock-Kurzweil integral from functions of signature $\mathbb R\to\mathbb R$ to functions of signature $\mathbb R^n \to \mathbb R$. One desirable property of such a generalisation would be a simple formulation of the Divergence Theorem, as well as a Fubini theorem. Would these difficulties still exist if we assumed the negation of the Axiom of Choice, and assumed instead the axiom that all subsets of $\mathbb R^n$ are Lebesgue measurable?
This builds on this question: What are the obstructions for a Henstock-Kurzweil integral in more than one dimension?
See also the Wikipedia entry on the Solovay model and the entry on dream mathematics at nLab.
