Tail Measure Upper Bound

Why

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.

Result

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

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