In a finite measure space, uniform integrability implies $L^p$ bounded?

Question

In the probability ambient we work with a measure space $(\Omega, \mathscr{F}, \mathbb{P})$ that is of finite measure (i.e. $\mathbb{P}(\Omega)<+\infty$).

I'm interest to understad all the relation between uniform integrable and $L^p$-bounded family.

Just for completeness i remember the main definitions:

UI:A family $(X_i)_{i \in \mathscr{I}}$ is called UI if

$ \sup_{i\in \mathscr{I}}\mathbb{E}(|X_i|\chi(|X_i|\geq K)) \to 0$ as $K\to+\infty$

$\mathbf{L^p}$-bounded :A family $(X_i)_{i \in \mathscr{I}}$ is called $L^p$-bounded if

$$\sup_{i\in \mathscr{I}}\mathbb{E}(|X_i|^p) < +\infty$$

So i know that:

1. UI implies $L^1$-bounded
2. $L^1$-bounded not implies UI
3. $L^p$-bounded implies UI when p>1 My question is: Is true the inverse implication of 3. ?

Answer 1

No, if $\mathscr{F}$ is uniformly integrable there may not exist a $p>1$ for which $\mathscr{F}$ is $L^p$ bounded. For example, if for all functions $f\in\mathscr{F}$ we have

$$\int_\Omega |f|\ln(1+|f|) \leq C $$

Then the family $\mathscr{F}$ is uniformly integrable.

In general, if $G:[0,\infty)\to [0,\infty)$ is increasing and grows superlinearly and we have $\int_\Omega G(|f|) \leq C$ for all $f\in\mathscr{F}$, then $\mathscr{F}$ is uniformly integrable, and the converse is also true: uniform integrability implies the existence of such a $G$. For a proof, see Bogachev Measure Theory, Theorem 4.5.9.