Distribution function (measure theory)


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.
Distribution functions are a generalization of distribution functions (in the sense of probability theory).

Definitions

The first definition presented here is typically used in Analysis to analysis properties of functions.
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).
It is well known result in measure theory 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, a few authors also refer to as a distribution function of the measure. That is:

Example

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

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