Definition (Uniformly Integrable)

Let $(f_{\alpha})_{\alpha\in\Lambda}\subset \mathscr{L}^{1}(\Omega,\mathcal{F},P)$ (i.e. a family of integrable functions); $(f_{\alpha})_{\alpha\in\Lambda}$ is called uniformly integrable if $$\lim_{ c \to \infty } \left(\sup_{\alpha\in\Lambda}\int\limits _{\{ |f_{\alpha}|\ge c\}}|f_{\alpha}| \, dP \right)=0$$