Distribution function (measure theory)


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 are a generalization of distribution functions.

Definition

Let be a measure on the real numbers, equipped with the Borel -algebra. Then the function
defined by
is called the distribution function of the measure.

Example

As the measure, choose the Lebesgue measure. Then by Definition of
Therefore, the distribution function of the Lebesgue measure is
for all

Comments

The definition of the distribution function differs slightly from the definition of the distribution function. The latter has the boundary conditions
This makes this distribution function well defined for all probability measures.
However, in the case of an unbounded measure, defining the distribution function as in probability theory by
can be without meaning. This is since many measures take on the value on all intervals, 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.