Vitali Theorems for Varying Measures

The classical Vitali theorem states that, under suitable assumptions, the limit of a sequence of integrals is equal to the integral of the limit functions. Here, we consider a Vitali-type theorem of the following form \int f_n dm_n → \int f dm for a sequence of pair ( f_n,m_n)_n and we study its asymptotic properties. The results are presented for scalar, vector and multivalued sequences of m_n-integrable functions f_n. The convergences obtained, in the vector and multivalued settings, are in the weak or in the strong sense for Pettis and McShane integrability. A list of known results on this topic is cited and new results are obtained when the ambient space Ω is not compact.