Let \(m\) be Lebesgue measure on \(\RR^n\), let \(f\colon\RR^n\to\CC\) be measurable, and let \(0<p<\infty\). For every \(\alpha>0\), \[ m\qty{\theset{x \in\RR^n \suchthat \abs{f(x)} \geq \alpha}} \leq \qty{\frac{\norm{f}_p}{\alpha}}^p, \qquad \norm{f}_p\coloneqq\qty{\int_{\RR^n}\abs{f(x)}^p\,dx}^{1/p}. \] Taking \(p=1\) gives \[ m\qty{\theset{x \in \RR^n\suchthat \abs{f(x)} \geq \alpha}} \leq \frac{1}{\alpha} \int_{\RR^n} \abs{f(x)} \, dx. \]
Let \(E_\alpha\coloneqq\theset{x \suchthat \abs{f(x)}\geq\alpha}\). Then \(\alpha^p\chi_{E_\alpha}\leq\abs{f}^p\) pointwise, and integrating gives \(\alpha^p\, m(E_\alpha)\leq\norm{f}_p^p\).
State Chebyshev's inequality.