Uniform integrability and Lebesgue convergence

Question

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?

Answer 1

1. There is no need of the parameter $n$: if $0\leq X\leq Y$ and $Y$ is integrable, it follows from the definition of Lebesgue integral that $X$ is integrable. If $0\leq s\leq X$ is a simple function, then $0\leq s\leq Y$ so $\sup\{\int S,0\leq S\leq X,S\mbox{ simple}\}$ is finite.

2. Try $X_n:=\sqrt n\chi_{((n+1)^{—1},n^{-1})}$.

3. Try $X_n:=n\chi_{((n+1)^{—1},n^{-1})}$.

4. Take $f$ an integrable function which is not in $L^2$. Then $X_n(x):=f(x+n)$ and $g(x)=x^2$.