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.

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One Response to Markov Inequality + PSLE One Dollar Question

  1. Pingback: f integrable implies set where f is infinite is measure zero | Singapore Maths Tuition

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