Fatou’s lemma and lower epi-limits of integral functionals

In this work, with the introduction in the σ -finite case of a modulus of equi-integrability, we prove some new extensions of Fatou’s lemma and some of its consequences in the convergence theory of integral functionals. We present the case of a sequence of integral functionals using an analog of Ioffe’s criterion for a sequence of integrands.