Markov's Inequality

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $f$ be an $A$-measurable function where $A \in \Sigma$.

Then:
 * $\displaystyle \mu \left({\left\{ {x \in A: \left\vert{f \left({x}\right)}\right\vert \ge t}\right\} }\right) \le \frac 1 t \int_A \left\vert{f}\right\vert \mathrm d \mu$

for any positive $t \in \R$.

Proof
Pick any $t$ and define:
 * $B = \left\{{x \in A: \left\vert{f \left({x}\right)}\right \vert \ge t}\right\}$

Let $\chi_B$ denote the indicator function of $B$ on $A$.

For any $x \in A$, either $x \in B$ or $x \notin B$.

In the first case:
 * $t \chi_B \left({x}\right) = t \cdot 1 = t \le \left\vert{f \left({x}\right)}\right\vert$

In the second case:
 * $t \chi_B \left({x}\right) = t \cdot 0 = 0 \le \left\vert{f \left({x}\right)}\right\vert$

Hence:
 * $\forall x \in A: t \chi_B \left({x}\right) \le \left\vert{f \left({x}\right)}\right\vert$

By Integral of Integrable Function is Monotone:
 * $\displaystyle \int_A t \chi_B \mathrm d \mu \le \int_A \left\vert{f}\right\vert \mathrm d \mu$

But by Integral of Integrable Function is Linear:
 * $\displaystyle \int_A t \chi_B \mathrm d \mu = t \int_A \chi_B \mathrm d \mu = t \mu \left({B}\right)$

Dividing through by $t$:
 * $\displaystyle \mu \left({B}\right) \le \frac 1 t \int_A \left\vert{f}\right\vert \mathrm d \mu$

Markov's Inequality in Probability
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

Markov's inequality asserts that for a random variable $X$:
 * $\Pr \left({\left\vert{X}\right\vert \ge t}\right) \le \dfrac {\mathrm E \left({\left\vert{X}\right\vert}\right)} t$

for any $t > 0$.

It can then be used to derive the probabilistic form of Chebyshev's Inequality.