As usual the convergence is denoted by $u_n \rightharpoonup u$ in $L^1(\Omega)$.
Definition (equiintegrability). For a family of integrable functions is equiintegrable if the following two conditions hold
- The set is tight, i.e. for any there exists a measurable set with
This condition is trivially true if .
- For any there exists such that for every measurable set with
Lemma (Equivalent characterisation of equiintegrability). Let , then is a family of equiintegrable functions if and only if
- the family is tight and
- there exists an increasing superlinear function such that
Theorem (Dunford-Pettis). A sequence converges weakly in if and only if
- the sequence is is equibounded in :
- and the sequence is equiintegrable.
Lemma (weak lower semicontinuity of convex functions). If is convex and
- K. H. Karlsen, “Notes on weak convergence (MAT4380 - Spring 2006).” pp. 1–14, 2006
- L. C. Evans."Weak convergence methods for nonlinear partial differential equations", volume 74 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1990.