Definition:Distribution Function of Finite Borel Measure

Definition

Let $\mu$ be a finite Borel measure on $\R$.

We define the distribution function of $\mu$, $F_\mu : \R \to \R$ by:

$\map {F_\mu} x = \map \mu {\hointl {-\infty} x}$

for each $x \in \R$.

Also see

• Definition:Cumulative Distribution Function - a notable special case where $\mu$ is the probability distribution of a real-valued random variable
• Definition:Distribution Function of Finite Signed Borel Measure - a generalisation of this concept to general finite signed Borel measures.

• Results about distribution functions of finite Borel measures can be found here.