Approximate Integrals and Approximate Variational Measures

Abstract

We establish that the approximate McShane integral is equivalent to the McShane integral and hence the Lebesgue integral for real functions. Then we consider a function V_{ap}^F analogous to V_F, for which some generalizations of the Hake’s theorem can be established. We also provide a measure theoretic characterization for the approximate Henstock integral, in terms of V_{ap}^F . We prove that V_{ap}^F (E) = V_{F}(E), for all measurable subsets E of the domain of F, if F is the primitive of an HK-integrable function. Some examples, when these two differ, are also provided.