Integration of functions ranging in complex Riesz space and some applications in harmonic analysis

We deal with complex Riesz spaces which can be obtained by the complexification of the usual Riesz space. We consider an extension of the Henstock-Kurzweil integral theory to complex Riesz spaces. Considering a derivation basis on zero-dimensional locally compact abelian group, we extend the Henstock-Kurzweil integration theory, with respect to this basis, to the case of functions with values in a complex Riesz space. We also investigate and solve the problem of recovering the primitive from a generalized derivative with respect to basis. Finally we apply this result to the problem of recovering coefficients of series with respect to characters in the compact case, and to the one of obtaining an inversion formula for multiplicative integral transforms, in the locally compact case.