Bounded Sequence in $L^p$ is Tight

Question

For $1 < p < \infty$, suppose that $\{f_n\}$ is bounded in $L^p(\mathbb{R})$. Is $\{fn\}$ tight?

We defined tight as : A family of measurable functions is said to be tight over $E$ provided for each $\epsilon >0$ there is a subset $E_0 \subseteq E$ of finite measure for which $\int_{E - E_0} |f| < \epsilon$

How do I prove this or a counterexample works? Any help. Thanks

Answer 1

"Tight convergence" excludes the possibility of mass escaping to $\infty$. Take as an example $f_n = \chi_{[n,n+1]}$. The sequence $\{f_n\}$ is bounded in $L^p$ but isn't tight. If it were, you would have $$m([n,n+1] \setminus E_0) = \int_{\mathbb R \setminus E_0} |f_n| < \epsilon$$ so that $$m([n,n+1] \cap E_0) > 1 - \epsilon$$ for all $n$. In this case $E_0$ couldn't have finite measure.