Let \(\mathcal{F}\) and \(\mathcal{G}\) be two families of uniformly integrable functions on an arbitrary measure space \((X, \mathscr{A}, \mu)\). Show that the following statements hold.

(i) Every finite family \(\left\{f_{1}, \ldots, f_{n}ight\} \subset \mathcal{L}^{1}(\mu)\) is uniformly integrable.

(ii) \(\mathcal{F} \cup\left\{f_{1}, \ldots, f_{n}ight\}, f_{1}, \ldots, f_{n} \in \mathcal{L}^{1}(\mu)\), is uniformly integrable.

(iii) \(\mathcal{F}+\mathcal{G}:=\{f+g: f \in \mathcal{F}, g \in \mathcal{G}\}\) is uniformly integrable.

(iv) c.h. \((\mathcal{F}):=\{t f+(1-t) \phi: f, \phi \in \mathcal{F}, 0 \leqslant t \leqslant 1\}\) ('c.h.' stands for convex hull) is uniformly integrable.

(v) The closure of c.h. \((\mathcal{F})\) in the space \(\mathcal{L}^{1}(\mu)\) is uniformly integrable.