Measure theory

The measure theory is a branch of mathematics that deals with the design and the study of measures employed. It is about generalizations of elementary geometric terms such as route length, surface area and volume to more complex quantities . Measure theory forms the foundation of modern integration and probability theory .

In measure theory, measure is understood to be a mapping that assigns real numbers to certain subsets of a basic set . For this purpose, the subsets must form a set system with certain properties and the assignment itself must also meet certain requirements. In practice, often only a partial assignment is known in advance. For example, one assigns the product of their edge lengths as area in the rectangle plane . Measure theory now examines on the one hand whether this assignment can be extended consistently and unambiguously to larger subset systems, and on the other hand whether additional desired properties are retained. In the example of the plane, one would of course also want to assign a meaningful area to circular disks and at the same time, in addition to the properties that are generally required of dimensions, translational invariance is also required, i.e. the content of a subset of the plane is independent of its position.

motivation

The complicated structure of the measure theory is caused by the fact that it is not possible to find a measure function that any arbitrary subset of the real number plane assigns a level which is the classic area corresponding to sensible. This attempt fails even with the one-dimensional number line, and it does not succeed even with higher dimensions. The question of whether this is possible was first formulated in 1902 by Henri Lebesgue in his Paris Thèse as a dimensional problem .

The following requirements are made of a meaningful correspondence of the area (to start from the 2-dimensional case):

1. A square with an edge length of one has an area of ​​one ( "normalized" ).
2. Moving, rotating or mirroring any surface does not change its surface area ( "motion invariance" ).
3. The area of ​​a finite or countably infinite union of pairwise disjoint areas is the sum of the areas of the partial areas ( σ-additivity ).

In 1905 Giuseppe Vitali was able to show that this problem cannot be solved for any subsets. It makes sense to weaken one of the requirements. If the third demand weakened and limited to finite unions, this leads to the content problem of Felix Hausdorff . Hausdorff was able to show in 1914 that this content problem in general (dimension greater than or equal to 3) cannot be solved. Exceptions are the real numbers and the real level, for which there is a solution to the content problem, a so-called content function (see definition of content ). However, if one restricts the quantities to be measured and only considers a certain system of subsets instead of arbitrary subsets, then one can generally solve the measurement problem for any spatial dimensions and define a dimension with the desired properties on this system of quantities (see definition of dimension ). It is then no longer necessary to restrict the σ-additivity requirement.

The measure theory deals with different set systems and the content functions that can be defined on them. Not only real set systems are considered, but abstract set systems on any basic set. This allows the results to be better applied in functional analysis and probability theory with little additional effort .

The property of additivity, which is central to the modern concept of measure , was introduced by Émile Borel in 1909 and was initially viewed with some criticism. In particular, it turns out that -additivity is such a strong requirement that not even the existence of an -additive function on the power set of an uncountable set is given, quite apart from additional requirements such as translational invariance ( Ulam's measure problem ). ${\ displaystyle \ sigma}$${\ displaystyle \ sigma}$${\ displaystyle \ sigma}$

The Jordanian construction also leads to only finitely additive contents, the finite additivity (a weaker property than additivity) is a consequence of the definition of the content. Borel, on the other hand, postulates the additivity of measure and thus determines the measures of sets which are contained in an algebra that is complete under countable applications of certain set operations . Henri Lebesgue's definition of the integral in 1902, however, retained the additivity. The restriction of additivity to a finite or countable number of sets can be seen as a way out of Zeno's (stylized) paradox of dimensions . ${\ displaystyle \ sigma}$${\ displaystyle \ sigma}$${\ displaystyle \ sigma}$${\ displaystyle \ sigma}$

Measure theory as the basis of the calculation of probability, as established by Kolmogorow , generally uses measures standardized to one as probabilities and not contents standardized to one. This is generally justified with the great technical advantages, as is the case with Kolmogorow. This is occasionally deviated from in subjectivist interpretations of probability, particularly prominently in Bruno de Finetti . On the other hand, there are Dutch-book arguments for the additivity of degrees of personal conviction ( english degree of belief ). ${\ displaystyle \ sigma}$

Definitions and examples

Hierarchy of the quantity systems used in measure theory

The quantities to be measured are summarized in quantity systems that are closed to different degrees in relation to quantity operations. Significant mass theoretical examples of set systems are:

Power set , σ-algebra , half-ring , ring , algebra , Dynkin system , monotonic classes or set system with constant mean stability .

The power set is the most comprehensive of all set systems and contains any subset of the basic set. The σ-algebra, which is the most important set system in measure theory, generally contains fewer sets than the power set.

Inclusions important for measure theory:

• Every power set is a σ-algebra and a Dynkin system.
• Every σ-algebra is an algebra.
• Every algebra is a ring.
• Each ring is a half ring.
• Each half ring is an average stable system of quantities.

Set functions such as content, premeasures, measures or external dimensions are defined on these set systems , which assign a value in (the extended positive real axis ) to each set of the set system . ${\ displaystyle [0, \ infty]}$

It should be noted that the terms mentioned (content, pre-measure, measure) are defined inconsistently in the literature, especially with regard to the underlying quantity system. For example, the term content is partially defined on a ring, half-ring or for any set of systems that contain the empty set. In the following, therefore, the general variant is given with reference to the consequences for the choice of special set systems.

content

Finite additivity for a content : The content of a finitely disjoint union is equal to the sum of the contents of the individual subsets.${\ displaystyle \ mu}$

A function which each lot from the lot system with more than a value maps, the in is called content , if for this mapping applies: ${\ displaystyle \ mu}$${\ displaystyle A}$${\ displaystyle {\ mathcal {C}}}$${\ displaystyle \ emptyset \ in {\ mathcal {C}}}$${\ displaystyle \ Omega}$${\ displaystyle \ mu (A)}$${\ displaystyle [0, \ infty]}$${\ displaystyle \ mu \ colon {\ mathcal {C}} \ rightarrow [0, \ infty]}$

• The empty set has zero value: .${\ displaystyle \ mu (\ emptyset) = 0}$
• The function is finite additive . So if there are finitely many pairwise disjoint sets from and , then we have${\ displaystyle A_ {1}, A_ {2}, \ dotsc, A_ {n}}$ ${\ displaystyle {\ mathcal {C}}}$${\ displaystyle \ textstyle \ bigcup _ {i = 1} ^ {n} A_ {i} \ in {\ mathcal {C}}}$
${\ displaystyle \ mu \ left (\ bigcup _ {i = 1} ^ {n} A_ {i} \ right) = \ sum _ {i = 1} ^ {n} {\ mu (A_ {i})} }$.

In particular, content can be expanded from half-rings to rings under certain circumstances.

Zero quantity

A set from is called a null set if holds. ${\ displaystyle A}$${\ displaystyle {\ mathcal {C}}}$${\ displaystyle \ mu (A) = 0}$

Premeasure

A σ-additive (or countable additive ) content is called a premeasure. If there is a content, then is a premeasure if for every sequence countably many pairwise disjoint sets from with holds: ${\ displaystyle \ mu \ colon {\ mathcal {C}} \ rightarrow [0, \ infty]}$${\ displaystyle \ mu}$${\ displaystyle (A_ {i}) _ {i \ in \ mathbb {N}}}$ ${\ displaystyle {\ mathcal {C}}}$${\ displaystyle \ textstyle \ bigcup _ {i = 1} ^ {\ infty} A_ {i} \ in {\ mathcal {C}}}$

${\ displaystyle \ mu \ left (\ bigcup _ {i = 1} ^ {\ infty} A_ {i} \ right) = \ sum _ {i = 1} ^ {\ infty} \ mu (A_ {i}) .}$

Pre-measures are particularly important for the Carathéodory extension set . It says that a premeasure can be continued to a measure on the algebra generated by the ring . If the premeasure is -finite , this continuation is clear. ${\ displaystyle \ sigma}$${\ displaystyle \ sigma}$

Measure

Countable additivity of a measure : The measure of a countable disjoint union is equal to the sum over the measures of the individual subsets.${\ displaystyle \ mu}$

Let be a function that assigns every set from the σ-algebra over a value in the set of extended real numbers (see below for possible generalizations ). A measure is called if the following conditions are met: ${\ displaystyle \ mu \ colon {\ mathcal {A}} \ rightarrow {\ overline {\ mathbb {R}}}}$${\ displaystyle A}$ ${\ displaystyle {\ mathcal {A}}}$${\ displaystyle \ Omega}$${\ displaystyle \ mu (A)}$${\ displaystyle {\ overline {\ mathbb {R}}}}$${\ displaystyle \ mu}$

• The empty set has zero dimension: .${\ displaystyle \ mu (\ emptyset) = 0}$
• Positivity: for everyone .${\ displaystyle \ mu (A) \ geq 0}$${\ displaystyle A \ in {\ mathcal {A}}}$
• The measure is countably additive (also σ-additive ): If there are countably many pairwise disjoint sets , then:${\ displaystyle A_ {1}, A_ {2}, A_ {3}, \ dotsc}$ ${\ displaystyle {\ mathcal {A}}}$
${\ displaystyle \ mu {} \ left (\ bigcup _ {k = 1} ^ {\ infty} A_ {k} \ right) = \ sum _ {k = 1} ^ {\ infty} {\ mu {} ( A_ {k})}}$.
Thus the measure is finitely additive by choosing the sequence of pairwise disjoint sets .${\ displaystyle A_ {1}, A_ {2}, \ dotsc, A_ {n} \ neq \ emptyset, A_ {n + 1}: = \ emptyset, \ dotsc}$${\ displaystyle {\ mathcal {A}}}$

Thus every measure is a premeasure over a σ-algebra, in particular all properties apply to contents and premeasures. Note that a measure of how the premeasure is defined in parts of the literature and the underlying Volume System with over is arbitrary. ${\ displaystyle {\ mathcal {C}}}$${\ displaystyle \ emptyset \ in {\ mathcal {C}}}$${\ displaystyle \ Omega}$

Measuring room, measurable quantities, measurable functions

Let be a σ-algebra made up of subsets of . Then the couple is called a measurable space or measurement room . The elements of are called measurable quantities . A function between two measuring rooms and is called measurable (more precisely - measurable) if the archetype of every measurable quantity is measurable. ${\ displaystyle {\ mathcal {A}}}$${\ displaystyle \ Omega}$${\ displaystyle (\ Omega, {\ mathcal {A}})}$${\ displaystyle A}$${\ displaystyle {\ mathcal {A}}}$${\ displaystyle f \ colon \ Omega \ to \ Omega '}$${\ displaystyle (\ Omega, {\ mathcal {A}})}$${\ displaystyle (\ Omega ', {\ mathcal {A}}')}$${\ displaystyle {\ mathcal {A}}}$${\ displaystyle {\ mathcal {A}} '}$

It should be noted that in measurement theory, on the one hand , there is talk of the measurability with regard to a measuring space and, on the other hand, of the measurability according to Carathéodory with regard to an external dimension . The latter can, however, be viewed equivalently as measurability with regard to the measurement space induced by the external dimension.

Dimensional space

A mathematical structure is called a measurement space if there is a measurement space and a measurement defined on this measurement space. An example of a measure space is the probability space from probability theory . It consists of the result set , the event algebra and the probability measure . ${\ displaystyle (\ Omega, {\ mathcal {A}}, \ mu)}$${\ displaystyle (\ Omega, {\ mathcal {A}})}$${\ displaystyle \ mu}$ ${\ displaystyle (\ Omega, {\ mathcal {A}}, P)}$ ${\ displaystyle \ Omega}$ ${\ displaystyle {\ mathcal {A}}}$ ${\ displaystyle P}$

Almost everywhere

A property applies almost everywhere (or -almost everywhere, or -most all elements) in if there is a null set such that all elements in the complement have the property. ${\ displaystyle \ mu}$${\ displaystyle \ mu}$${\ displaystyle (\ Omega, {\ mathcal {A}}, \ mu)}$

Note that the set of all for which the property does not apply does not necessarily have to be measurable, but only has to be contained in a measurable set of measure zero. ${\ displaystyle \ omega \ in \ Omega}$

In stochastics , the property is also referred to almost everywhere on the probability space as an almost certain (or almost certain ) property. ${\ displaystyle (\ Omega, {\ mathcal {A}}, P)}$ ${\ displaystyle P}$

completion

Subsets of zero sets are called negligible . A measure space is called complete if all negligible quantities are measurable. Let it denote the set of all negligible sets. ${\ displaystyle {\ mathcal {N}}}$

The triple is called the completion of , if one puts: (where the symmetrical difference is) and . ${\ displaystyle (\ Omega, {\ mathcal {A}} ', \ mu')}$${\ displaystyle (\ Omega, {\ mathcal {A}}, \ mu)}$${\ displaystyle {\ mathcal {A}} ': = \ {A \ bigtriangleup N: A \ in {\ mathcal {A}}, N \ in {\ mathcal {N}} \}}$${\ displaystyle \ bigtriangleup}$${\ displaystyle \ mu '(A \ bigtriangleup N): = \ mu (A)}$

Examples

• The zero dimension that assigns the value to every quantity .${\ displaystyle A}$${\ displaystyle \ mu (A) = 0}$
• An example of a content is the Jordan content, which can be used to define the multi-dimensional Riemann integral .
• The counting measure assigns each subset of a finite or countable infinite set the number of its elements, .${\ displaystyle A}$${\ displaystyle \ mu (A) = | A |}$
• The Lebesgue measure on the set of real numbers with Borel's σ-algebra , defined as a translation-invariant measure with .${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mu (\ left [0,1 \ right]) = 1}$
• The hair measure on locally compact groups .
• A probability measure or normalized measure is a measure with .${\ displaystyle \ mu (\ Omega) = 1}$
• The counting measure on the set of natural numbers is infinite, but σ-finite.${\ displaystyle \ mathbb {N}}$
• The canonical Lebesgue measure on the set of real numbers is also infinite, but σ-finite, because it can be represented as a union of countably many finite intervals .${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ left [k, k + 1 \ right]}$

Generalizations

A possible generalization concerns the value range of the function . ${\ displaystyle \ mu}$

Another possibility of generalization is the definition of a measure on the power set.

Results

The set of Hadwiger classified all possible dimensions invariable in : the Lebesgue measure is also a special case as the Euler characteristic . There are also connections to the Minkowski functionals and the cross-measures . ${\ displaystyle \ mathbb {R} ^ {n}}$