Let \((X, \mathscr{A}, \mu)\) be an arbitrary measure space. Show that a family \(\mathcal{F} \subset \mathcal{L}^{1}(\mu)\) is uniformly integrable if, and only if, the following condition holds:

\[\forall \epsilon>0 \quad \exists g_{\epsilon} \in \mathcal{L}_{+}^{1}(\mu): \sup _{u \in \mathcal{F}} \int\left(|u|-g_{\epsilon} \wedge|u|ight) d \mu<\epsilon\]

Give a simplified version of this equivalence for finite measure spaces.