# Density function

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}$

## Individual evidence

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