Density function

This article deals with the general concept of the density function in measure theory. For the special case in stochastics see probability density function . For physical density functions, see density curve and mass distribution .

A density function , or density for short , is a special real-valued function that occurs mainly in the mathematical sub-areas of stochastics and measure theory . There, density functions are used to construct dimensions or signed dimensions using integrals .

The best-known example of density functions are the probability density functions from probability theory . With their help, many probability measures can be constructed without having to resort to more deeply-based methods and structures.

definition

Let a dimension space and a positive, quasi-integrable function be given ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle \mu }$

${\displaystyle f\colon X\to \mathbb {R} }$.

Then you can get through

${\displaystyle \mu _{f}(A):=\int _{A}f(x)\,\mathrm {d} \mu (x)}$ for all ${\displaystyle A\in {\mathcal {A}}}$

define a measure. The function is then called the density function of the measure. ${\displaystyle f}$

Are reversed and dimensions on and is ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle (X,{\mathcal {A}})}$

${\displaystyle \nu (A)=\int _{A}f_{\nu }(x)\,\mathrm {d} \mu (x)}$for a positive quasi-integrable function and all , ${\displaystyle f_{\nu }}$ ${\displaystyle A\in {\mathcal {A}}}$

this is the name of the density function of measure with respect to measure . The function is then also referred to as the Radon-Nikodým density or Radon-Nikodým derivative and noted as being. ${\displaystyle f_{\nu }}$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle {\tfrac {\mathrm {d} \nu }{\mathrm {d} \mu }}}$

The definition for signed dimensions is identical in both cases, only the positivity of the quasi-integrable functions is dropped.

Examples

Probability density functions

Typical examples of density functions are probability density functions . These are density functions with respect to the Lebesgue measure or the Lebesgue integral , in which the measure of the base space is one. Specifying such a function is a simple way of using probability measures ${\displaystyle \lambda }$ ${\displaystyle f_{P}}$

${\displaystyle P(A)=\int _{A}f_{P}(x)\,\mathrm {d} \lambda (x)}$

define. Probability measures that can be defined in this way are called absolutely continuous probability measures . They allow an elementary access to probability theory, often the Lebesgue integral is not used and the Riemann integral is used instead . Then the notation is used instead of . ${\displaystyle \mathrm {d} x}$ ${\displaystyle \mathrm {d} \lambda (x)}$

Counting densities

Counting densities , also called probability functions, are another example of density functions . In the simplest case, you assign a positive number to every natural number:

${\displaystyle f\colon \mathbb {N} \to [0,\infty )}$.

The function values ​​add up to one and thus over define

${\displaystyle P(\{k\}):=f(k)}$

a discrete probability distribution . If we choose as a measure the counting measure on , so ${\displaystyle \mu }$ ${\displaystyle \mathbb {N} }$

${\displaystyle P(A)=\sum _{k\in A}f(k)=\int _{A}f(k)\,\mathrm {d} \mu (k)}$.

Counting densities are therefore density functions with regard to the counting measure.

existence

By definition, every positive quasi-integrable function can be used in combination with a measure to define a further measure and thus be declared a density function.

If, however, two measures are given, the question arises whether a density function has or vice versa. Radon-Nikodým's theorem answers this question : ${\displaystyle \mu ,\nu }$ ${\displaystyle \nu }$ ${\displaystyle \mu }$

If σ-finite and is absolutely continuous with respect to , then has a density function with respect to . ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle \mu }$

See also

• Density bundle

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .

Individual evidence

1. ↑ Klenke: Probability Theory. 2013, p. 159.