Uniform Integrability - different characterisation - prove (ii)

Question

Probability with Martingales:

---

---

For the 'only if' part

assuming the hint is true, then I guess

we have

1. $\forall \varepsilon_1 > 0, \exists K \ge 0$ s.t. $$E[|X|1_{|X| > K}] < \varepsilon_1$$

2. $$E[X1_F] \le E[|X|1_{|X| > K} + KP(F)] \le \varepsilon_1 + K\delta = \varepsilon_1 + K(\frac{\varepsilon - \varepsilon_1}{K}) = \varepsilon$$

Is that right?

---

Actually, why do we even need the hint?

If a random variable is uniformly integrable, then it is integrable. Hence by this lemma

---

---

we actually have (ii) already?

Answer 1

$\mathcal{C}$ is UI $\Rightarrow$ $(i)$

Let $\epsilon>0$. Then, since $\mathcal{C}$ UI there exists $K>0$ s.t. \begin{equation} E[|X|] = E[|X|1_{|X|>K}+|X|1_{|X|\le K}]\le \epsilon + K \end{equation} for all $X\in\mathcal{C}$. Thus $(i)$.

$\mathcal{C}$ is UI $\Rightarrow$ $(ii)$

You want to find for a given $\epsilon>0$ a $\delta>0$ such that for all $F\in\mathcal{F}$ with $P(F)<\delta$ and all $X\in\mathcal{C}$ we have \begin{equation} E[|X|\mid F]<\epsilon. \end{equation} It follows directly from the hint. Since $\mathcal{C}$ is UI choose $K>0$ such that \begin{equation} E[|X|1_{|X|>K}]<\frac{\epsilon}{2}, \end{equation} for every $X\in\mathcal{C}$. Choose $\delta<\frac{\epsilon}{2K}$ and you are done.

You cannot use the lemma, because it gives a particular $\delta$ which depends on $X$ every time, but you need a "general" $\delta$ which works for each $X\in\mathcal{C}$.