Let $(X,\mathfrak{X},\mu)$ be a measure space and let $f \in L^p(\mu)$. Show that \begin{equation} \mu(|f| \geq n)\rightarrow 0 \end{equation}
as $n \rightarrow \infty$.
Since $f \in L^p(\mu)$, we know that: \begin{equation*} ||f||_{p} = \left(\int_X |f|^p \,d\mu\right)^{1/p} < \infty \end{equation*}
It is clear that if $m
as \begin{equation} |f(x)|\leq m \Rightarrow |f(x)|\leq n \text{ whenever } m
Furthermore, we know that: \begin{equation} (||f||_{p})^p = \int_X |f|^p \,d\mu < \infty \end{equation}
However, I am unsure of how to proceed from here. Since $p$ is fixed, is it sufficient to claim thus, that there is some upper bound $M$ such that $|f(x)|\leq M$ for all $x\in X$ and hence when $n>M$ we have that $\{x \in X : |f(x)| \leq n \} = \varnothing$ which being empty, has measure zero?