Uniform Integrability
# Definition 1
Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let ${X_{t}}{t \in T}$ be a collection of Random Variables with Index Set $T$. Then we say that ${X{t}}{t \in T}$ is uniformly integrable if $$\lim\limits{a \to 0} \sup\limits_{t \in T} \int\limits_{|X_{t}| > a} |X_{t}| d \mathbb{P} = 0$$