Let $\mathcal{R}$ be a family of random variables on the common probability space $(\Omega, \mathcal{F}, \mathbb{P})$. What is meant by saying that $\mathcal{R}$ is uniformly integrable? Explain the use of uniform integrability in the study of convergence in probability and in $L^{1}$. [Clear definitions should be given of any terms used, but proofs may be omitted.]

Let $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ be uniformly integrable families of random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. Show that the family $\mathcal{R}$ given by

$$\mathcal{R}=\left\{X+Y: X \in \mathcal{R}_{1}, Y \in \mathcal{R}_{2}\right\}$$

is uniformly integrable.