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Suppose $\mu$ is a finite measure and for some $\gamma > 0$, we have$$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty.$$Does it follow that $\{f_n\}$ is uniformly integrable?

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Yes, if $\mu$ is a finite measure and the family $\{f_n\}$ is bounded in $L^p$ for some $p>1$ then it is uniformly integrable.

To prove this, let $M=\sup_{n}||f_n||_p$ and let $q$ be the conjugate exponent of $p$ (note that $q<\infty$). Given $\varepsilon>0$, let $\delta=\frac{\epsilon^q}{M^q}$, and suppose that $\mu(E)<\delta$. Then by Holder's inequality, $$ \int_E|f_n|\;d\mu\leq ||f_n||_p||1_E||_q\leq M\mu(E)^{\frac{1}{q}}<\varepsilon $$ for all $n$, hence $$ \sup_n\int_E|f_n|\;d\mu<\varepsilon $$

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