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 ∫fndmn→∫fdm for a sequence of pair (fn,mn)n and we study its asymptotic properties. The results are presented for scalar, vector and multivalued sequences of mn-integrable functions fn. 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.

Keywords

• setwise convergence
• Weak and Vague convergence of measures
• Vitali theorem
• uniform integrability
• equi-integrability
• Pettis integral