Chebyshev's inequality is the following:
Suppose $X, \mu$ is a measure space, and $f \in L^p(X, \mu)$, then for all $t > 0$
$\mu( {x \in X : |f(x)| \geq t } ) \leq \frac{1}{t^p} \|f\|_{L^p(X, \mu)}^p$.
The proof is trivial:
Observe that
$\mu( {x \in X : |f(x)| \geq t } )t^p = \int_{X} 1_{|f| \geq t}(x)t^p \leq \int_{X} |f|^p = \|f\|_{L^p(X, \mu)}^p$
and divide both sides by $t^p$.
This is a fundamental inequality in the the study of the interpolation of L^p spaces.
