Vitali Theorems for Varying Measures

Abstract

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 \[\int f_n\,dm_n \rightarrow \int f \,dm\] 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 $\Omega$ is not compact.