In mathematics, in particular in measure theory, there are different notions of distribution function and it is important to understand the context in which they are used (properties of functions, or properties of measures).
Distribution functions (in the sense of measure theory) are a generalization of distribution functions (in the sense of probability theory).
The first definition[1] presented here is typically used in Analysis (harmonic analysis, Fourier Analysis, and integration theory in general) to analysis properties of functions.
provides information about the size of a measurable function
The next definitions of distribution function are straight generalizations of the notion of distribution functions (in the sense of probability theory).
It is well known result in measure theory[2] that if
is a nondecreasing right continuous function, then the function
defined on the collection of finite intervals of the form
μ
extends uniquely to a measure
μ
that included the Borel sets.
induce the same measure, i.e.
μ
μ
is a measure on Borel subsets of the real line that is finite on compact sets, then the function
is a nondecreasing right-continuous function with
This particular distribution function is well defined whether
is finite or infinite; for this reason,[3] a few authors also refer to
as a distribution function of the measure
That is: As the measure, choose the Lebesgue measure
Therefore, the distribution function of the Lebesgue measure is