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.
Definition 1: Suppose
is a
measure space, and let
be a real-valued
measurable function. The distribution function associated with
is the function
given by
It is convenient also to define
.
The function 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).
Definition 2. Let
be a finite
measure on the space
of
real numbers, equipped with the
Borel -algebra. The
distribution function associated to
is the function
defined by
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 by
extends uniquely to a measure on a -algebra that included the Borel sets. Furthermore, if two such functions and induce the same measure, i.e. , then is constant. Conversely, if is a measure on Borel subsets of the real line that is finite on compact sets, then the function defined by
is a nondecreasing right-continuous function with such that .
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:
Definition 3: Given the measure space
, if
is finite on compact sets, then the nondecreasing right-continuous function
with
such that
is called the
canonical distribution function associated to
.