As is well-known, the space of square-integrable functions (say, on $[0,\,1]$) where the integral is a Riemann integral is not complete. If one completes it, one obtains the $L^{2}([0,\,1])$ Hilbert space. On the other hand, the space of square-integrable functions where the integral is a Lebesgue integral is complete, and is in fact equal to that same $L^{2}([0,\,1])$ Hilbert space we just mentioned.
Now, what about the space of square-integrable functions where the integral is the Henstock–Kurzweil (or, gauge) integral? Is it complete? If yes, is it a Hilbert space? If not, can we complete it to get a Hilbert space? Has this Hilbert space been studied?
