Uniform integrability

Summary

A family of integrable functions is uniformly integrable when the part of each function above a large level carries uniformly little integral. It prevents mass from escaping into tall spikes, in the same way for every member of the family. On a finite measure space it is exactly the condition that upgrades convergence almost everywhere or in measure to convergence in $L^1$. It plays a central role in probability, for instance in martingale convergence.