----- > [!proposition] Proposition. ([[Markov's inequality]]) > Suppose $(X, \Sigma, \mu)$ is a [[measure|measure space]] [[Lp-norm|and]] $h \in \mathcal{L}^{1}(\mu)$. Then $\mu(\{ x \in X : |h(x)| \geq c \}) \leq \frac{1}{c} \|h\|_{1}$ > for every $c>0$. ^proposition > [!proof]- Proof. ([[Markov's inequality]]) > Suppose $c>0$. Then $\begin{align} \mu(\{ x \in X : |h(x)| \geq c\}) &= \frac{1}{c} \int_{\{ x \in X: |h(x) \geq c| \}} c \, d\mu \\ & \leq \frac{1}{c} \int _{\{ x \in X: |h(x)| \geq c \}} |h| \, d\mu \\ & \leq \frac{1}{c} \|h\|_{1}. \end{align}$ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```