Solution
= Solution
The assertion is \b[false]. On $([0,1],\mathcal B,\text{Lebesgue measure})$, put $X_n=n\mathbf1_{(0,1/n)}$. Then $\mathbb E|X_n|=1$ for every $n$, so the family is $L^1$ bounded. For every $M$, choose $n>M$; then $\mathbb E[|X_n|\mathbf1_{|X_n|>M}]=1$. The tail criterion for <uniform integrability> fails.