Measurable function

Measurable functions are examined in measure theory , a sub-area of mathematics that deals with the generalization of length and volume concepts. It is required of measurable functions that the archetype of certain quantities lies in a certain system of quantities . Measurable functions play an important role in stochastics and measure theory, as they can be used to construct random variables and image measures .

definition

Given are two measurement spaces and , that is, a basic set and a σ-algebra on this set, as well as a function ${\displaystyle (X_{1},{\mathcal {A}}_{1})}$ ${\displaystyle (X_{2},{\mathcal {A}}_{2})}$

${\displaystyle f\colon X_{1}\to X_{2}}$.

The function is now called a measurable function if the archetype of every set under element is off. ${\displaystyle f}$ ${\displaystyle A_{2}\in {\mathcal {A}}_{2}}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {A}}_{1}}$

This condition is formalized

${\displaystyle f^{-1}(A_{2})\in {\mathcal {A}}_{1}}$, for everyone . ${\displaystyle A_{2}\in {\mathcal {A}}_{2}}$

Such a function is also referred to as - - measurable . A function is called Borel-measurable ( Lebesgue-measurable ) if it can be measured with respect to two Borel σ-algebras (Lebesgue σ-algebras). Mixed forms ( Lebesgue-Borel-measurable or Borel-Lebesgue-measurable ) are sometimes used. It should be noted that no dimension needs to be defined in order to define a measurable function. ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{2}}$

Elementary examples

• If two measurement spaces and are given, and if the trivial σ-algebra is, then every function is - -measurable, regardless of the choice of the function and the σ-algebra . This is because always and applies. However, these sets are always contained in σ-algebra . However, is chosen as σ-algebra the power set , then also each function - -measurable, independent of the choice of the function and the σ-algebra . This is due to the fact that every archetype always lies in the power set, since by definition this contains every subset of the superset. ${\displaystyle (X_{1},{\mathcal {A}}_{1})}$ ${\displaystyle (X_{2},{\mathcal {A}}_{2})}$ ${\displaystyle {\mathcal {A}}_{2}=\{\emptyset ,X_{2}\}}$ ${\displaystyle f\colon X_{1}\to X_{2}}$ ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle f^{-1}(\emptyset )=\emptyset }$ ${\displaystyle f^{-1}(X_{2})=X_{1}}$ ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{1}={\mathcal {P}}(X_{1})}$ ${\displaystyle f\colon X_{1}\to X_{2}}$ ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle f^{-1}(A)}$
• Every constant function, i.e. every function of the form for everyone , is measurable. Namely is so is ${\displaystyle f(x_{1})=c}$ ${\displaystyle x_{1}\in X_{1}}$ ${\displaystyle A\in {\mathcal {A}}_{2}}$

${\displaystyle f^{-1}(A)={\begin{cases}X_{1}&{\text{ falls }}c\in A\\\emptyset &{\text{ falls }}c\notin A\end{cases}}}$

Since the basic set and the empty set are contained in any σ-algebra, they are in particular contained in and the function is measurable. ${\displaystyle {\mathcal {A}}_{1}}$

• Are and measuring rooms, then for any the indicator function one - -measurable function. Then and as well as and . However, according to the assumption, these sets are contained in σ-algebra. ${\displaystyle (X_{1},{\mathcal {A}}_{1})}$ ${\displaystyle (\{0,1\},{\mathcal {P}}(\{0,1\}))}$ ${\displaystyle M\in {\mathcal {A}}_{1}}$ ${\displaystyle f=\chi _{M}}$ ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {P}}(\{0,1\})}$ ${\displaystyle f^{-1}(\{1\})=M}$ ${\displaystyle f^{-1}(\{0\})=X_{1}\setminus M}$ ${\displaystyle f^{-1}(\emptyset )=\emptyset }$ ${\displaystyle f^{-1}(\{0,1\})=X_{1}}$

classification

Archetype of a measurable amount

The concept of measurability is motivated by the definition of integration by Henri Lebesgue : For the Lebesgue integration of a function with regard to the Lebesgue measure , a measure must be assigned to quantities of the form . Examples of functions for which this is not possible are indicator functions of Vitali sets . The definition of the Lebesgue integration for any dimension spaces then leads to the above definition of the measurable function. ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }$ ${\displaystyle f^{-1}([a,b])}$

The concept of the measurable function has parallels to the definition of the continuous function . A function between topological spaces and is continuous if and only if the archetypes of open sets of are open sets of . The σ-algebra generated by the open sets is the Borelian σ-algebra . A continuous function is thus measurable with respect to the Borel σ-algebras of and borel-measurable for short . Lusin's theorem is a certain inversion of this statement . ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{2},}$

As random variables, measurable functions play an important role in probability theory .

properties

Measurable functions and generating systems

Often a σ-algebra is much too large to be able to specify any set directly from it or to check the archetype of each set. Checking a function for measurability is made easier by the fact that this only has to be done on the prototypes of a producer. If, therefore, is a producer of , i.e. is , then the function is measurable if and only if ${\displaystyle {\mathcal {E}}_{2}}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle \sigma ({\mathcal {E}}_{2})={\mathcal {A}}_{2}}$ ${\displaystyle f}$

${\displaystyle f^{-1}(E)\in {\mathcal {A}}_{1}}$

applies to all . ${\displaystyle E\in {\mathcal {E}}_{2}}$

It follows directly from this that continuous functions between topological spaces which are provided with Borel's σ-algebra are always measurable, since archetypes of open sets are always open. Since the Borelian σ-algebra is generated from the open sets and therefore the archetypes of the generator are again in the generator, measurability follows.

Initial σ algebra

For each figure , where the σ-algebra is provided, a smallest σ-algebra can be given, with respect to which the function can be measured. This σ-algebra is then called the initial σ-algebra of the function and denotes it with or with . It can also be defined for any family of functions . It is then the smallest σ-algebra with respect to which all are measurable and is then denoted by or . Due to the operational stability of the archetype, the initial σ-algebra is already for a single function . ${\displaystyle f\colon X_{1}\to X_{2}}$ ${\displaystyle X_{2}}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle f}$ ${\displaystyle \sigma (f)}$ ${\displaystyle {\mathcal {I}}(f)}$ ${\displaystyle (f_{i})_{i\in I}}$ ${\displaystyle f_{i}}$ ${\displaystyle \sigma (f_{i}\colon i\in I)}$ ${\displaystyle {\mathcal {I}}(f_{i}\colon i\in I)}$ ${\displaystyle f}$ ${\displaystyle f^{-1}({\mathcal {A}}_{2})}$

Chaining of measurable functions

If , and measuring rooms and is - -measurable and - -measurable, then the function - -measurable. Under certain circumstances, the measurability of linked functions can also be used to determine the measurability of their sub-functions: If functions from to and is the initial σ-algebra, then a function from to is measurable if and only if - is measurable for all . ${\displaystyle (X_{1},{\mathcal {A}}_{1})}$ ${\displaystyle (X_{2},{\mathcal {A}}_{2})}$ ${\displaystyle (X_{3},{\mathcal {A}}_{3})}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle g}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle {\mathcal {A}}_{3}}$ ${\displaystyle h(x)=g(f(x))}$ ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{3}}$ ${\displaystyle f_{i}}$ ${\displaystyle (X_{2},{\mathcal {A}}_{2})}$ ${\displaystyle (X_{i},{\mathcal {A}}_{i})}$ ${\displaystyle {\mathcal {A}}_{2}={\mathcal {I}}(f_{i})}$ ${\displaystyle f}$ ${\displaystyle (X_{1},{\mathcal {A}}_{1})}$ ${\displaystyle (X_{2},{\mathcal {A}}_{2})}$ ${\displaystyle f_{i}\circ f}$ ${\displaystyle i\in I}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{i}}$

Factoring lemma

The factorization lemma is a measure theoretical auxiliary theorem about the measurability of functions that is used in some far-reaching stochastic constructions and theorems of mathematical statistics . The lemma is used, for example, in the construction of the factorized conditional expectation , which is a step on the way to the regular conditional distribution . The lemma says: If a figure

${\displaystyle f\colon \Omega _{1}\to \Omega _{2}}$

is given, then is the figure

${\displaystyle g\colon \Omega _{1}\to {\overline {\mathbb {R} }}}$

if and -measurable if a function exists so that ${\displaystyle \sigma (f)-{\mathcal {B}}({\overline {\mathbb {R} }})}$ ${\displaystyle \varphi }$

${\displaystyle \varphi \colon (\Omega _{2},{\mathcal {A}}_{2})\to ({\overline {\mathbb {R} }},{\mathcal {B}}({\overline {\mathbb {R} }}))}$

is measurable and

${\displaystyle g=\varphi \circ f}$

applies. There is a σ-algebra on and the σ-algebra generated by . ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle \Omega _{2}}$ ${\displaystyle \sigma (f)}$ ${\displaystyle f}$

Measurability of real-valued functions

Verification

For a mapping of a measuring room according to , it is possible to measure exactly when one of the quantity systems ${\displaystyle f}$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$ ${\displaystyle f}$

• ${\displaystyle \{f\leq a\},a\in \mathbb {R} }$,
• ${\displaystyle \{f<a\},a\in \mathbb {R} }$,
• ${\displaystyle \{f\geq a\},a\in \mathbb {R} }$,
• ${\displaystyle \{f>a\},a\in \mathbb {R} }$

in lies. Here etc. is to be understood as an abbreviation for . It would also be sufficient if only all rational numbers were run through, because the specified intervals always form a generating system of Borel's σ-algebra. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \{f\leq a\}}$ ${\displaystyle \{x\in X\mid f(x)\leq a\}=f^{-1}((-\infty ,a])}$ ${\displaystyle a}$

Measurable functions

For example, the following functions can be measured: ${\displaystyle g\colon (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))\to (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

${\displaystyle g(x)=\max(0,x)=:x^{+}}$

${\displaystyle g(x)=\min(0,x)=:x^{-}}$

${\displaystyle g(x)=|x|}$

${\displaystyle g(x)=\operatorname {sgn} (x)}$

If a function is also given, then it can be measured precisely when each of its component functions is - -measurable. ${\displaystyle f\colon (X_{1},{\mathcal {A}}_{1})\to (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}))}$ ${\displaystyle f_{i}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {B}}(\mathbb {R} )}$

Are measurable functions by , so are and measurable. Is measurable from to , so is measurable. If you agree on the convention , it can even be measured. ${\displaystyle f,g}$ ${\displaystyle (X_{1},{\mathcal {A}}_{1})}$ ${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}))}$ ${\displaystyle f+g}$ ${\displaystyle f-g}$ ${\displaystyle h}$ ${\displaystyle (X_{1},{\mathcal {A}}_{1})}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$ ${\displaystyle f\cdot h}$ ${\displaystyle {\tfrac {x}{0}}=0}$ ${\displaystyle {\tfrac {f}{h}}}$

If a sequence of functions - measurable functions is given, the infimum, the supremum as well as the limes inferior and the limes superior of this sequence can be measured again. ${\displaystyle (X_{1},{\mathcal {A}}_{1})}$ ${\displaystyle ({\overline {\mathbb {R} }},{\mathcal {B}}({\overline {\mathbb {R} }}))}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$

Approximation

Every positive measurable function can be approximated by a monotonically increasing function sequence of simple functions (i.e. linear combinations of indicator functions of measurable quantities). A sequence of functions that does this is, for example

${\displaystyle f_{n}(x)=\min(2^{-n}\lfloor 2^{n}f(x)\rfloor ;n)}$.

This approximation property is used in the construction of the Lebesgue integral , which is initially only defined for simple functions and is then continued for all measurable functions.

Lebesgue and Borel measurable functions

A (real) Lebesgue-Borel measurable function is not necessarily Borel-Borel measurable. A Lebesgue-Borel measurable function is also not necessarily Lebesgue-Lebesgue measurable. The concatenation of two Lebesgue-Borel measurable functions is not necessarily Lebesgue-Borel measurable.

Related terms

Strong measurability

Is a function in a metric space point-wise limit of elementary functions , i.e. H. measurable functions with a finite image, it is called “strongly measurable” .

• Every measurable function with a separable image is highly measurable.
• Every highly measurable function is measurable.

Strong measurability and measurability only differ from one another if the target space is not separable. This is the case, for example, with the definition of generalized integrals such as the Bochner integral .

Bimeasurable functions

Measurable functions whose inverse mapping is also measurable are called bimeasurable functions .

Demarcation

A subset of a measurement space is called measurable if it is an element of the σ-algebra of the measurement space and a measure can thus potentially be assigned to it. Furthermore, according to Carathéodory, there is still the measurability of quantities with regard to an external measure . Only an external dimension is required here.

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 .
• Henri Lebesgue: Leçons sur l'intégration et la recherche des fonctions primitives. Gauthier-Villars, Paris 1904.

Individual evidence

1. ↑ and are each measurable quantities based on and . ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle (X_{1},{\mathcal {A}}_{1})}$ ${\displaystyle (X_{2},{\mathcal {A}}_{2})}$
2. ^ Robert B. Ash, Catherine Doléans-Dade: Probability and measure theory. 2nd edition. Academic Press, San Diego CA et al. a. 2000, ISBN 0-12-065202-1 , p. 41.
3. ↑ Vladimir I. Bogachev: Measure theory. Volume 1. Springer, Berlin a. a. 2007, ISBN 978-354-03451-3-8 , p. 193.