Is this true for any $L^p$ space?

Question

Suppose $f\in L^p$ with $1\leq p<\infty$. Let $E_\alpha=\{x\mid|f(x)|>\alpha\}$. Then $$\lim_{\alpha\to\infty}\int_{E_\alpha}|f|^p d\mu=0$$ Any hints would be appreciated.

Answer 1

Yes, it is true. Since \begin{equation*} \lVert f \rVert_p^p = \sum_{i = 0}^\infty \int_{E_{i} \setminus E_{i+1}} \lvert f\rvert^p d\mu \end{equation*} is finite, the remainder terms \begin{equation*} \int_{E_n} \lvert f\rvert^p d\mu = \sum_{i = n}^\infty \int_{E_{i} \setminus E_{i+1}} \lvert f\rvert^p d\mu \end{equation*} of the series have to converge to zero.