Uniform Integrability - sufficient condition and bounded convergence theorem with weaker hypothesis

Question

Probability with Martingales:

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1. How does the result follow? Do we choose $K = (\frac{\varepsilon}{A})^{\frac{1}{1-p}}1_{A \ne 0}$

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2. Why do we have that inequality?

Answer 1

1. Yes, we can choose such a $K$, or just notice that $$\lim_{K\to +\infty}K^{1-p}A=0$$ and that the bound of $\mathbb E\left(\left|X\right|;\left|X\right|\gt K\right)$ is uniform with respect to $X\in\mathcal C$.

2. The follows from the inclusion $$\left\{\left|X\right|\gt K+k^{-1}\right\}\subset \left\{\left|X-X_n\right|\gt k^{-1}\right\}.$$ To see this, it suffices to show that $$\left\{\left|X-X_n\right|\leqslant k^{-1}\right\}\subset \left\{\left|X\right|\leqslant K+k^{-1}\right\}.$$ If $\omega$ is such that $\left|X(\omega)-X_n(\omega)\right|\leqslant k^{-1}$, then by assuption, $$\left|X(\omega)\right|\leqslant \left|X_n(\omega)-X(\omega)\right|+\left|X(\omega)\right|\leqslant k^{-1}+K.$$