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Distribution function (measure theory)

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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 byIt 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.