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

then

### References

- 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.