Let $(X,\mathcal A,\mu)$ be a measurable space. A sequence $\{f_n\}$ of positive measurable functions is said to be uniformly integrable if for any $\epsilon$, there exists a $\delta$ such that if $U$ is a measurable subset with $\mu(U)<\delta$, then $\int_U f_n d\mu<\epsilon$ for any $n$.
Now assume that $\{f_n\}$ is any sequence of positive measurable functions. Under which conditions can we extract a uniformly integrable subsequences from it? Is there any known theorem that ensures that?
Uniform integrability of $(f_n)$ is equivalent to the existence of an increasing convex function $\phi$ mapping $[0,+\infty[$ onto $[0,+\infty[$ such that the sequence integrals of $\phi(|f_n|)$ are bounded.
Hence a criterion for the existence of a uniformly integral subsequence is the existence of a positive increasing convex function $\phi$ mapping $[0,+\infty[$ onto $[0,+\infty[$ such that $\liminf_n \int_X \phi(|f_n|) d\mu < +\infty$.