Uniform Integrability - different characterisation - prove hint

Question

Probability with Martingales:

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For the 'only if' part

how to prove the hint? i'm guessing it's something to do with

$$E[X 1_F] \le E[X1_{\Omega}]$$

$$= E[X 1_{|X| > K}] + E[X 1_{|X| \le K}]$$

$$ \le E[X 1_{|X| > K}] + K E[1_{|X| \le K}]$$

$$\le E[X 1_{|X| > K}] + K P(|X| \le K)$$

Answer 1

But with you attempt, you do not reach the conclusion. Let $K>0$, then we have \begin{equation} E[|X|1_{F}] = E[|X|1_F1_{|X|>K}+|X|1_F1_{|X|\le K}]\le E[|X|1_{|X|>K}]+E[K1_F]= E[|X|1_{|X|>K}]+KP(F). \end{equation}