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 (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).

Definitions

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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  .

Example

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

Comments

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  • The distribution function   of a real-valued measurable function   on a measure space   is a monotone nonincreasing function, and it is supported on  . If   for some  , then  
  • When the underlying measure   on   is finite, the distribution function   in Definition 3 differs slightly from the standard definition of the distribution function   (in the sense of probability theory) as given by Definition 2 in that for the former,   while for the latter,  
  • When the objects of interest are measures in  , Definition 3 is more useful for infinite measures. This is the case because   for all  , which renders the notion in Definition 2 useless.

References

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  1. ^ Rudin, Walter (1987). Real and Complex Analysis. NY: McGraw-Hill. p. 172.
  2. ^ Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications. NY: Wiley Interscience Series, Wiley & Sons. pp. 33–35.
  3. ^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 164. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.