Approximate integrals and approximate variational measures

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_Fᵃᵖ 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_Fᵃᵖ . We prove that V_Fᵃᵖ(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.