In the classical Fourier analysis, the representation of the double Fourier transform as the integral of is usually defined from the Lebesgue integral. Using an improper Kurzweil–Henstock integral, we obtain a similar representation of the Fourier transform of non-Lebesgue integrable functions on . We prove the Riemann–Lebesgue lemma and the pointwise continuity for the classical Fourier transform on a subspace of non-Lebesgue integrable functions which is characterized by the bounded variation functions in the sense of Hardy–Krause. Moreover, we generalize some properties of the classical Fourier transform defined on , where , yielding a generalization of the results obtained by E. Hewitt and K. A. Ross. With our integral, we define the space of integrable functions which contains a subspace whose completion is isometrically isomorphic to the space of integrable distributions on the plane as defined by E. Talvila [The continuous primitive integral in the plane, Real Anal. Exchange 45 (2020), 2, 283–326]. A question arises about the dual space of the new space .