Definition:Distribution Function of Finite Signed Borel Measure

Definition

Let $\mu$ be a finite signed 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 Borel Measure - an instantiation where $\mu$ is more specifically a finite measure.

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

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.4$: Functions of Finite Variation