## Markov Inequality + PSLE One Dollar Question

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Markov inequality is a useful inequality that gives a rough upper bound of the measure of a set in terms of an integral. The precise statement is: Let $f$ be a nonnegative measurable function on $\Omega$. The Markov inequality states that for all $K>0$, $\displaystyle \mu\{x\in\Omega:f(x)\geq K\}\leq\frac{1}{K}\int fd\mu$.

The proof is rather neat and short. Let $E_K:=\{x\in\Omega: f(x)\geq K\}$ Then,

\begin{aligned} \int f d\mu &\geq \int_{E_K} fd\mu\\ &\geq \int_{E_K}K d\mu\\ &=K \mu(E_K) \end{aligned}

Therefore, $\mu(E_K)\leq\frac{1}{K}\int fd\mu$.