Quantitative mapping

A set- valued mapping (also called a set- valued function ) is a special mapping in mathematics in which the elements of the target space are sets. They are used, for example, in game theory and statistics . In contrast, a set function is usually a function whose definition set is a set system .

definition

Let be a system of sets over the basic set , so . Then a figure is called ${\displaystyle {\mathcal {M}}}$ ${\displaystyle W}$ ${\displaystyle {\mathcal {M}}\subseteq {\mathcal {P}}(W)}$

${\displaystyle \varphi \colon V\to {\mathcal {M}}}$

a set- valued mapping . The definition set is arbitrary. The elements of the target set are again sets. ${\displaystyle V}$ ${\displaystyle {\mathcal {M}}}$

Examples

• The mapping defined by is a quantitative mapping. ${\displaystyle \varphi \colon \mathbb {R} \to {\mathcal {P}}(\mathbb {R} )}$ ${\displaystyle \varphi (x)=[0,x]}$
• If there is a function, then the mapping that assigns the archetype to each is a set-valued mapping . ${\displaystyle f\colon D\to Z}$ ${\displaystyle y\in Z}$ ${\displaystyle f^{-1}(\{y\})}$ ${\displaystyle Z\to {\mathcal {P}}(D)}$
• More generally, every set family and thus every set sequence is a set- valued mapping.

use

Set-valued functions are used in game theory, for example, to guarantee the well-defined definition of the arg max function. This gives its maximum places for a given function. They are also used in statistics, where they are used to determine confidence intervals with the help of range estimates. Range estimators are set-valued functions that assign a set (usually an interval, a sphere or an ellipse) to every observation of a statistical experiment. If you choose these sets and the corresponding estimators appropriately, you then get a confidence interval for the appropriate level of error.

literature

• Jean-Pierre Aubin , Hélène Frankowska : Set-Valued Analysis . Birkhauser Verlag , Boston, ISBN 978-0-8176-4847-3 .