# weak L¹-convergence

 Let , a sequence  converges weakly to  if

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

1. The set  is tight, i.e. for any  there exists a measurable set  with 

This condition is trivially true if .
2. 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

1. the family  is tight and
2. there exists an increasing superlinear function  such that

Theorem (Dunford-Pettis). A sequence  converges weakly in  if and only if

1. the sequence is  is equibounded in :

2. and the sequence  is equiintegrable.

Lemma (weak lower semicontinuity of convex functions). If  is convex and

then

### References

1. K. H. Karlsen, “Notes on weak convergence (MAT4380 - Spring 2006).” pp. 1–14, 2006
2. 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.