ThmDex – An index of mathematical definitions, results, and conjectures.
Formulation F10944 on R2535: Chebyshov's inequality
F10944
Formulation 3
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) $f : X \to [0, \infty]$ is an D5610: Unsigned basic Borel function on $M$
Let $\varphi : [0, \infty] \to \mathbb{R}$ be a D5321: Standard-isotone basic real function such that
(i) $\lambda > 0$ is a D993: Real number
(ii) \begin{equation} \varphi(\lambda) \neq 0 \end{equation}
Then \begin{equation} \mu (f \geq \lambda) \leq \frac{1}{\varphi(\lambda)} \int_X \varphi(f(x)) \, \mu(d x) \end{equation}