A). Given that $ |X_n| \leq Y $ and $Y \in L$. Try to show $X_n$ is lebesgue integrable.
b). Try to give any example for which $X_n \longrightarrow^{L} X$ yet $\not\exists Y \in L$ with $|X_n| \leq Y$.
c). Try to give any example for which $X_n \to X$ w.p.1, and $X_n, X \in L$ yet $X_n \not\longrightarrow^L X$.
d). If $X_n$ is uniform integrable, does it follow that $g(X_n)$ is uniform integrable if g is continuous?