Let $C_1$ and $C_2$ be uniformly integrable collections of functions on a measure space $(\Omega,\mathscr{F},\mu)$, and let $C$ be the collection defined as \begin{equation*} C = \{f_1 + f_2:f_1\in C_1, f_2\in C_2\}. \end{equation*} How to show that $C$ is uniformly integrable?
Rogers & Williams state this (as Lemma 20.7, the proof is left as an exercise) for random variables. I presume it also holds for functions on general measure spaces.
My attempt so far, using the definition of uniform integrability by Hunt:
Let $\varepsilon > 0$. Since $C_1$ and $C_2$ are uniformly integrable, there are integrable $g_1\geq 0$ and $g_2\geq 0$ such that \begin{equation*} \sup_{f_1\in C_1} \int_{|f_1| > g_1}|f_1|d\mu < \varepsilon \quad\text{ and }\quad \sup_{f_2\in C_2} \int_{|f_2| > g_2}|f_2|d\mu < \varepsilon . \end{equation*} Now I want to use these bounds to bound $\sup_{f_1+f_2\in C}\int_{|f_1+f_2| > g}|f_1+f_2|d\mu$, where I need to choose an integrable $g\geq 0$: for any $f_1\in C_1$ and $f_2\in C_2$ and any integrable $g\geq 0$, if $|f_1 + f_2| > g$ then \begin{equation*} |f_1| + |f_2| \geq |f_1 + f_2| > g, \end{equation*} hence \begin{equation*} \int_{|f_1 + f_2| > g}|f_1 + f_2|d\mu \leq \int_{|f_1| + |f_2| > g}|f_1|d\mu + \int_{|f_1| + |f_2| > g}|f_2|d\mu . \end{equation*} While on the right-hand side I have integrals of $|f_1|$ and $|f_2|$ separately, they are over the set where $|f_1| + |f_2| > g$, which I cannot bound by integrals of $|f_1|$ over the set $|f_1| > g_1$ respectively $|f_2|$ over the set $|f_2| > g_2$ through some appropriate choice of $g$.
I feel like I am overlooking something. Any help is appreciated!