Singular function

A singular function is a special real function in measure theory . Singular functions are characterized by seemingly contradicting properties. So they are steady and constant almost everywhere , but growing at the same time . So the growth takes place on a set of volume zero.

Singular functions occur, for example, in the Lebesgue decomposition of functions or as distribution functions of continuously singular probability distributions .

definition

Let the Lebesgue-Borel measure be on and ${\ displaystyle \ beta}$ ${\ displaystyle \ mathbb {R}}$

{\ displaystyle F \ colon \ mathbb {R} \ supset I \ to \ mathbb {R}}

a real function on an interval . ${\ displaystyle I}$

Then a singular function is called if and only if it is continuous and increasing and its derivative - is zero almost everywhere . ${\ displaystyle F}$ ${\ displaystyle F '}$ ${\ displaystyle \ beta}$

Comment, characteristics and example

Plot of the Cantor function (10 iterations)

The function from the definition does not have to be differentiable. It follows automatically from the monotony that it is differentiable almost everywhere. ${\ displaystyle F}$ ${\ displaystyle \ beta}$

Singular functions are closely related to singular measures : is a singular function if and only if the associated Lebesgue-Stieltjes measure is atomless and singular with respect to the Lebesgue-Borel measure. ${\ displaystyle F}$

The standard example of a singular function is the Cantor function . An approximation of the Cantor function is shown on the right, a detailed construction can be found in the main article. It is noteworthy that it is locally constant on the complement of the Cantor set . ${\ displaystyle {\ mathcal {C}}}$

Web links

BI Golubov: Singular function . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

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 .