(a) Suppose that $\mathcal{X}=\left(X_{n}\right)$ is a sequence of random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Give the definition of what it means for $\mathcal{X}$ to be uniformly integrable.

(b) State and prove Hölder's inequality.

(c) Explain what it means for a family of random variables to be $L^{p}$ bounded. Prove that an $L^{p}$ bounded sequence is uniformly integrable provided $p>1$.

(d) Prove or disprove: every sequence which is $L^{1}$ bounded is uniformly integrable.