Convergence for varying measures

Some limit theorems of the type ${\int }_{\mathrm{\Omega }}{f}_{n}d{m}_n---->{\int }_{\mathrm{\Omega }}fdm$ are presented for scalar, (vector), (multi)-valued 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.