In mathematics, a distribution function is a real function in measure theory. From every measure on the algebra of Borel sets of real numbers, a distribution function can be constructed, which reflects some of the properties of this measure. Distribution functions (in the sense of measure theory) are a generalization of distribution functions (in the sense of probability theory).

Definition

Let ${\displaystyle \mu }$ be a measure on the real numbers, equipped with the Borel ${\displaystyle \sigma }$-algebra. Then the function

${\displaystyle F_{\mu }\colon \mathbb {R} \to \mathbb {R} \cup \{+\infty ,-\infty \}}$

defined by

${\displaystyle F_{\mu }(t)={\begin{cases}\mu ((0,t])&{\text{if }}t\geq 0\\-\mu ((t,0])&{\text{if }}t<0\end{cases}}}$

is called the (right continuous) distribution function of the measure ${\displaystyle \mu }$.^[1]

Example

As the measure, choose the Lebesgue measure ${\displaystyle \lambda }$. Then by Definition of ${\displaystyle \lambda }$

${\displaystyle \lambda ((0,t])=t-0=t{\text{ and }}-\lambda ((t,0])=-(0-t)=t}$

Therefore, the distribution function of the Lebesgue measure is

${\displaystyle F_{\lambda }(t)=t}$

for all ${\displaystyle t\in \mathbb {R} }$

Comments

The definition of the distribution function (in the sense of measure theory) differs slightly from the definition of the distribution function (in the sense of probability theory). The latter has the boundary conditions

${\displaystyle \lim _{t\to -\infty }F(t)=0{\text{ and }}\lim _{t\to \infty }F(t)=1.}$

This makes this distribution function well defined for all probability measures. However, in the case of an unbounded measure ${\displaystyle \mu }$, defining the distribution function as in probability theory by

${\displaystyle F_{\mu }(t)=\mu ((-\infty ,t])}$

can be without meaning. This is since many measures take on the value ${\displaystyle +\infty }$ on all intervals ${\displaystyle (-\infty ,t]}$, making their distribution function a constant function with value infinity. This is for example the case for the Lebesgue measure. To avoid this pathological case, the distribution function is defined to be zero at the origin. This makes sure that even for unbounded measures, the distribution function is well defined and finite close to the origin.

References