On the size of the set of non-measurable Riemann integrable functions

Source abstract

We establish the existence of an infinite-dimensional linear subspace consisting of functions (taking values in certain Banach spaces) that are bounded and Riemann integrable but non-measurable. We also show existence of an infinite-dimensional linear subspace consisting of bounded functions which are Riemann integrable and measurable but Darboux non-integrable, with values in certain Banach spaces.