An Upper Bound for the Convergence of Integral Functionals

Given a $\sigma$-finite measure space $(\Omega, \mathcal{T}, \mu)$ endowed with a $\mu$-complete tribe, a separable Banach space $E$, we consider a topological vector space $(X,T)$, $X$ being a decomposable subspace of measurable $E$-valued functions defined on $\Omega$. Under a reasonable assumption on the vector topology $T$, we show that if ${(f_{n})}_{n}$ is a sequence of extended real-valued measurable integrands defined on the product $\Omega\times E$, with upper epi-limit (or upper $\Gamma$-limit) $f=ls_{e} f_{n}$, then $I_f$ is in many cases an upper bound for the $T$-upper epi-limit of the sequence $(I_{f_{n}})_n$, where $I_f$, $I_{f_{n}}$ are the integral functionals defined on $X$ associated to the integrands $f$, $f_n$. The cases of Lebesgue spaces endowed with its strong, weak, or Mackey topologies are reached. We discuss also the necessity of the given conditions.