In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales.

Uniform integrability is an extension to the notion of a family of functions being dominated in ${\displaystyle L_{1}}$ which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition:

Definition A: Let ${\displaystyle (X,{\mathfrak {M}},\mu )}$ be a positive measure space. A set ${\displaystyle \Phi \subset L^{1}(\mu )}$ is called uniformly integrable if ${\displaystyle \sup _{f\in \Phi }\|f\|_{L_{1}(\mu )}<\infty }$, and to each ${\displaystyle \varepsilon >0}$ there corresponds a ${\displaystyle \delta >0}$ such that

whenever ${\displaystyle f\in \Phi }$ and ${\displaystyle \mu (E)<\delta .}$

Definition A is rather restrictive for infinite measure spaces. A more general definition of uniform integrability that works well in general measure spaces was introduced by G. A. Hunt.

Definition H: Let ${\displaystyle (X,{\mathfrak {M}},\mu )}$ be a positive measure space. A set ${\displaystyle \Phi \subset L^{1}(\mu )}$ is called uniformly integrable if and only if

where ${\displaystyle L_{+}^{1}(\mu )=\{g\in L^{1}(\mu ):g\geq 0\}}$.

Since Hunt's definition is equivalent to Definition A when the underlying measure space is finite (see Theorem 2 below), Definition H is widely adopted in Mathematics.