Definition 10

Let $X$ be a locally compact space, $\mu$ a measure on $X$, $F$ a Banach space, and $p \in [1, +\infty[$. A subset $H$ of $\mathscr{L}_F^p(X, \mu)$ is said to be equi-integrable of order $p$ (for $\mu$) if it satisfies the following conditions:

(i) For every $\varepsilon > 0$ there exists a $\delta > 0$ such that, for every integrable set $A$ of measure $|\mu|(A) \leq \delta$ and every $f \in H$,

(ii) For every $\varepsilon > 0$ there exists a compact subset $K$ of $X$ such that, for every $f \in H$, $\int |f|^p \varphi_{X-K} d|\mu| \leq \varepsilon$.

When $p = 1$ we say 'equi-integrable' instead of 'equi-integrable of order 1'.