Set function

In mathematics , set functions are functions that assign values to certain sets (the sets of a set system ) , usually non-negative real numbers or the value . On the other hand, functions that take on sets as values ​​are called set-valued functions . ${\displaystyle \infty }$

Set functions form the basis for measure theory , where, among other things, set functions such as measures or contents are examined for more precise properties.

motivation

Set functions are particularly important in measure theory . The idea of ​​measure theory is to be able to assign a (real) measure to quantities . A simple example would be to count the elements of a finite set: The set then has measure 4. ${\displaystyle \{1,3,5,27\}\subseteq \mathbb {N} }$

However, you don't want to assign a value to just a set, but to an entire system of sets , i.e. a set of sets. For example, if you look at the set system:, and define a function that counts the number of elements, you get a set function. For the set function then applies , , , . ${\displaystyle {\mathcal {C}}:=\{\{3\},\{5\},\{1,3,27\},\{1,3,5,27\}\}\subseteq {\mathcal {P}}(\mathbb {N} )}$ ${\displaystyle f\colon {\mathcal {C}}\to \mathbb {R} }$ ${\displaystyle f}$ ${\displaystyle f(\{3\})=1}$ ${\displaystyle f(\{5\})=1}$ ${\displaystyle f(\{1,3,27\})=3}$ ${\displaystyle f(\{1,3,5,27\})=4}$

Now one can examine set functions for their properties. In measure theory, one often demands certain stability properties, such as additivity , that is, if one divides a set, the two new sets together must have the same value as the initial set. This is true in the above example when counting, so is . ${\displaystyle f(\{5\})+f(\{1,3,27\})=1+3=4=f(\{1,3,5,27\})}$

Formal definition

Let be a non-empty set and a set system with . Next, first , briefly . Then they call each image with a set function . ${\displaystyle X}$ ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {P}}(X)}$ ${\displaystyle \emptyset \in {\mathcal {C}}}$ ${\displaystyle W=\mathbb {R} ^{+}\cup \{\infty \}}$ ${\displaystyle W=[0,\infty ]}$
${\displaystyle f\colon \,{\mathcal {C}}\to W}$ ${\displaystyle f(\emptyset )=0}$

One usually speaks of a quantity function if or is ( signed measure ) or ( complex measure ). ${\displaystyle W=\mathbb {R} \cup \{-\infty \}}$ ${\displaystyle W=\mathbb {R} \cup \{\infty \}}$ ${\displaystyle W=\mathbb {C} }$

Examples

• Certain point sets of the plane (the areas ) can be assigned an area as a measure . This assignment is (like the previous one) always greater than or equal to 0 and σ-additive; such a set function is called a measure .
• In analysis, the area between the x-axis and a function graph is determined using the integral. Areas below the x-axis are given a negative sign . This assignment is also σ-additive; such a quantity function is called a signed measure .
• Probability measures are σ-additive set functions that assume values ​​between 0 and 1 and assign measure 1 to the entire base set ("certain event").
• An outer measure is a σ-subadditive set function that is always greater than or equal to 0. This can be achieved, for example, by assigning to each subset of the level the infimum of the areas of all supersets that can be measured as areas. Usually one proceeds the other way around and constructs an external measure in order to obtain a measure by suitable restriction of the measurable quantities (e.g. construction of the Lebesgue measure ).

Special properties of set functions

The set function f is called:

General properties

monotonic if for ${\displaystyle A\subseteq B\Rightarrow f(A)\leq f(B)}$ ${\displaystyle A,B\in {\mathcal {C}}}$

finally , if for everyone ${\displaystyle A\in {\mathcal {C}}\Rightarrow f(A)<\infty }$

σ-finite , if there is a sequencewithandfor all. ${\displaystyle (A_{j})_{j\in \mathbb {N} }}$ ${\displaystyle \bigcup _{j\in \mathbb {N} }A_{j}=\Omega }$ ${\displaystyle f(A_{j})<\infty }$ ${\displaystyle j\in \mathbb {N} }$

limited if for all : ${\displaystyle A\in {\mathcal {C}}}$ ${\displaystyle \sup _{A\in {\mathcal {C}}}|f(A)|<\infty }$

complete , if for all with and : applies. ${\displaystyle A\in {\mathcal {C}}}$ ${\displaystyle f(A)=0}$ ${\displaystyle B\subseteq A}$ ${\displaystyle B\in {\mathcal {C}}}$

Compatibility of addition and union

additive , if for disjoint sets from with ${\displaystyle f(A\cup B)=f(A)+f(B)}$ ${\displaystyle A,B}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle A\cup B\in {\mathcal {C}}}$

finally additive if for any pairs disjoint sets out ${\displaystyle f(\bigcup _{j=1}^{m}A_{j})=\sum _{j=1}^{m}f(A_{j})}$ ${\displaystyle A_{1},...,A_{m}}$ ${\displaystyle {\mathcal {C}}}$

σ-additive (sigma-additive), iffor every sequence of disjoint setsinwith ${\displaystyle f(\bigcup _{j\in \mathbb {N} }A_{j})=\sum _{j\in \mathbb {N} }f(A_{j})}$ ${\displaystyle (A_{j})_{j\in \mathbb {N} }}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \bigcup _{\in \mathbb {N} }A_{j}\in {\mathcal {C}}}$

subadditive if for off with ${\displaystyle f(A\cup B)\leq f(A)+f(B)}$ ${\displaystyle A,B}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle A\cup B\in C}$

finitely subadditive , if for all sets off with ${\displaystyle f(\bigcup _{j=1}^{m}A_{j})\leq \sum _{j=1}^{m}f(A_{j})}$ ${\displaystyle A_{1},...,A_{m}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \bigcup _{j=1}^{m}A_{j}\in C}$

σ-subadditive (sigma-subadditive), iffor every sequence of setsinwith ${\displaystyle f(\bigcup _{j\in \mathbb {N} }A_{j})\leq \sum _{j\in \mathbb {N} }f(A_{j})}$ ${\displaystyle (A_{j})_{j\in \mathbb {N} }}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \bigcup _{j\in \mathbb {N} }A_{j}\in {\mathcal {C}}}$

subtractive if for all with , and : . In doing so, one demands in order to avoid undefined differences . ${\displaystyle A,B\in {\mathcal {C}}}$ ${\displaystyle B\subseteq A}$ ${\displaystyle f(B)<\infty }$ ${\displaystyle A\setminus B\in {\mathcal {C}}}$ ${\displaystyle f(A\setminus B)=f(A)-f(B)}$ ${\displaystyle f(B)<\infty }$ ${\displaystyle \infty -\infty }$

modular , if for everyone and : ${\displaystyle A,B\in {\mathcal {C}}}$ ${\displaystyle A\cup B,A\cap B\in {\mathcal {C}}}$ ${\displaystyle f(A\cup B)+f(A\cap B)=f(A)+f(B)}$

continuity

→ Main article : σ-continuity

continuously from below , if for every monotonically increasing sequence with and : ${\displaystyle (A_{j})_{j\in \mathbb {N} }}$ ${\displaystyle A_{j}\in {\mathcal {C}}}$ ${\displaystyle \bigcup _{j\in \mathbb {N} }A_{j}\in {\mathcal {C}}}$

${\displaystyle f\left(\bigcup _{i\in \mathbb {N} }A_{i}\right)=\sup _{i\in \mathbb {N} }f(A_{i})}$

applies.

continuously from above , if for every monotonically decreasing sequence with , and : ${\displaystyle (A_{j})_{j\in \mathbb {N} }}$ ${\displaystyle A_{j}\in {\mathcal {C}}}$ ${\displaystyle f(A_{1})<\infty }$ ${\displaystyle \bigcap _{j\in \mathbb {N} }A_{j}\in {\mathcal {C}}}$

${\displaystyle f\left(\bigcap _{i\in \mathbb {N} }A_{i}\right)=\inf _{i\in \mathbb {N} }f(A_{i})}$

applies.

${\displaystyle \emptyset }$-continuously from above , if for every monotonically decreasing sequence with , and : ${\displaystyle (A_{j})_{j\in \mathbb {N} }}$ ${\displaystyle A_{j}\in {\mathcal {C}}}$ ${\displaystyle f(A_{1})<\infty }$ ${\displaystyle \bigcap _{j\in \mathbb {N} }A_{j}=\emptyset }$

${\displaystyle \inf _{i\in \mathbb {N} }f(A_{i})=0}$

applies.

Relationships between properties

• Every σ-additive set function is finitely additive and every finitely additive set function is additive.
• Every finite set function is σ-finite.
• Every additive set function is subtractive.
• Every bounded set function is finite.
• If a ring is a ring , then every additive set function is finitely additive and every subadditive set function is finitely subadditive. ${\displaystyle {\mathcal {C}}}$

literature

• Herbert Amann, Joachim Escher : Analysis. Volume 3. 2nd edition, Birkhäuser, Basel et al. 2008, ISBN 978-3-7643-8884-3 .
• Klaus D. Schmidt: Measure and Probability. Springer, Berlin et al. 2009, ISBN 978-3-540-89729-3 .