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Real-Valued Random Variable Variance
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Tail Measure Upper Bound


Consider bounding the measure where two functions differ. We would look at their absolute value, and the measure of where this is greater than zero. The absolute value of their difference is a non-negative measurable random variable.


We bound the measure that a non-negative measurable real-valued function exceeds some value by its integral.

Suppose $g: X \to \R _+$ is measurable on $(X, \mathcal{A} , \mu )$. Then

\[ \mu (\Set*{x \in X}{g(x) \geq t}) \leq \frac{\int g d\mu }{t} \quad \text{ for all } t > 0 \]

Let $A = \Set*{x \in X}{g(x) \geq t}$. Define $h: X \to R$ by

\[ h(x) = \begin{cases} 1 & \text{ if } g(x) \geq t \\ 0 & \text{ otherwise. } \end{cases} \]

First, $\mu (A) = \int h d\mu $. Second, $h \leq g$. So:

\[ \mu (\Set*{x \in X}{g(x) \geq t}) = \int h d\mu \leq \int g/t d\mu = \frac{\int g d\mu }{t}. \]

This proposition is universally referred to as Markov's inequality (or the Markov inequality).1

  1. We eschew this terminology in accordance with the project's policy on naming. ↩︎
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