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 $\int f_{n}d{m}_n\to \int fdm$ 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 $\Omega$ is not compact.