Let $Q=(0,T)\times\Omega.$ I am completely lost with this:
No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a sequence being equiintegrable implies convergence in $L^1$. (Let us just take $p=p'=2$). Here we can assume that $u_\delta \in L^2(Q)$ if need be.
Source is this paper on page 426.