Uniform integrability

Uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. The definition used in measure theory is closely related to, but not identical to, the definition typically used in probability.

Textbooks on real analysis and measure theory often use the following definition.

Let ${\displaystyle (X,{\mathfrak {M}},\mu )}$ be a positive measure space. A set ${\displaystyle \Phi \subset L^{1}(\mu )}$ is called uniformly integrable if to each ${\displaystyle \epsilon >0}$ there corresponds a ${\displaystyle \delta >0}$ such that

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