**Definition (equiintegrability).**For \(\Omega \subset \mathbb{R}^n\) a family of integrable functions \(\mathcal{U}\subset L^1(\Omega)\) is equiintegrable if the following two conditions hold

- The set \(\mathcal{U}\) is tight, i.e. for any \(\varepsilon > 0\) there exists a measurable set \(A\) with \(|A|<\infty\) \[ \forall u\in \mathcal{U}: \quad \int_{\Omega\backslash A} | u | < \varepsilon. \] This condition is trivially true if \(|\Omega| < \infty\).
- For any \(\varepsilon>0\) there exists \(\delta >0\) such that for every measurable set \(E\) with \(|E|\leq \delta\) \[ \forall u\in\mathcal{U}: \quad \int_E | u | \;\dx{x} < \varepsilon. \]

**Lemma (Equivalent characterisation of equiintegrability).**Let \(\Omega\subset \R^n\), then \(\mathcal{U}\subset L^1(\Omega)\) is a family of equiintegrable functions if and only if

- the family \(\mathcal{U}\) is tight and
- there exists an increasing superlinear function \(\Psi: [0,\infty)\to [0,\infty]\) such that \[ \sup_{u\in \mathcal{U}} \int_\Omega \Psi(|u|) \; \dx{x} < \infty . \]

**Theorem (Dunford-Pettis).**A sequence \((u_n)_{n\in \mathbb{N}} \subset L^1(\Omega)\) converges weakly in \(L^1(\Omega)\) if and only if

- the sequence is \(u_n\) is equibounded in \(L^1(\Omega)\): \[ \sup_n \Vert u_n \Vert_{L^1(\Omega)} < \infty . \]
- and the sequence \(u_n\) is equiintegrable.

**Lemma (weak lower semicontinuity of convex functions).**If \(F:\R \to \R\) is convex and \[ u_n \rightharpoonup u \quad \text{in } L^1(\Omega). \] then \[ \int F(u) \;\dx{x} \leq \liminf_{n\to \infty} \int F(u_n) \;\dx{x} . \]

### 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.