the masstheorie is a subsection of mathematics, which makes the elementary-geometrical terms for route distance possible , area , volume generalized and it to assign to quantities also more complicated a measure. It forms the foundation of the modern integration and probability theory.

As measure one understands in the masstheoriean allocation from real or complex numbers to a subset system over a fundamental set. The allocation and the subset system are to possess thereby certain characteristics. In practice frequently only a partial allocation is from the beginning well-known. For example one arranges the product in the level rectanglestheir edge lengths as area too. The masstheorie examines now on the one hand whether in consistent way and clearly this allocation can be extended to larger subset systems and on the other hand, whether thereby additional desired characteristics remain. In the example of the level one would like naturally also circular disks a meaningful areaassign and apart from the characteristics, which one completely generally requires of masses, also translation invariance will at the same time demand, i.e. contents of a subset of the level are unabhänging from its position.

Table of contents

definitions and examples

Messraum, measurable quantities

of an accurate definition of the fundamental ideas of the masstheorie begin we with a fundamental set Ω. If a certain quantity of Σfrom subsets of Ω an σ-algebra forms, then is called each quantity, the element of Σ is measurable , (English. measurable), and the fundamental set Ω with the structure Σ is called Messraum (English. measurable space). A function, which receives the structure of a Messraums, is called measurable function.


The demand that Σ is an σ-algebra, means
  • that Σ with each quantity of S contains also their complement Ω \ of S
  • that Σ contains the empty quantity (and concomitantly their complement Ω), and
  • that Σ is final concerning the countable combination.

Examples ofMessräume:

  • Each finite or countable infinite quantity, in particular thus also the quantity of the natural numbers of N, forms a Messraum with its power quantity as σ-algebra.
  • If A is a subset of Ω, then is {Ω, <math> \ empty </math>, A, Ω \ A} an σ-algebra.

Measure, measure area

a measure μ is a function, which assigns a value to each quantity of S from Σ μ (S). This value is either a nonnegative real number or <math> \ infty< /math> (see below because of possible Verallgemeinerungen). Furthermore must apply:

  • The empty quantity has thatMeasure of zero: <math> \ mu (\ empty) = 0< /math>.
  • The measure is countable additive (also σ-additive), i.e., if E 1, E 2, E 3,… many disjunkte quantities in pairs from Σ are countable and E their set union are, then is the measure μ (E) equal the sum <math> \ sum {\ mu {} (E_k)}</math>.

The structure (Ω, Σ, μ) of a Messraums as well as one on this defined measure is called measure area (English. measure space).

Examples of mass:

empty set, completely, nearly everywhere

an empty set is a quantity of S from Σ with the measure <math> \ mu (s) = 0< /math>. A measure means completely, if each subset is contained of each empty set in Σ. A characteristicapplies nearly everywhere in Ω, if it does not apply only in an empty set.

Examples of empty sets:

  • The empty quantity is always an empty set.
  • Each at the most countable subset of the real numbers is an empty set concerning. the Lebesgue measure.

finite, σ-finite

a measure is called finally, if <math> \ mu (\ omega) < \ infty< /math>. A measure means σ-finally, if Ω the combination of a countable consequence of measurable quantities <left> math \ \ {S_1, S_2, S_3, \ dots \ right \}< /math> , all this is a finite measure <math> \ mu {} (S_k) < \ infty< /math> have.

some beautiful characteristics, the certain analogy have σ-finite mass toothe characteristics separabler topological area exhibit.


  • the counting measure on the quantity <math> \ N< /math> the natural numbers is infinite, but σ-finite.
  • The canonical Lebesgue measure on the quantity <math> \ R< /math> the real numbers is likewise infinite, but σ-finite, because <math> \ R< /math> can as combination of countable manyfinite intervals <math> \ [k, k+1 \ right] /math< left> are represented.


a possible Verallgemeinerung concerns the range of values of the function μ.

  • One can permit negative real or complex values (complex or marked measure).
  • A further example of a Verallgemeinerung is the spectral measure, its values linear operators are. This measure is used in particular in the Funktionalanalysis for the spectral theorem.

Another possibility of the Verallgemeinerung is the definition of a measure on the power quantity.

to historical first additives mass were finally introduced. The modern definition, to that-according to a measure countable to additiveis, proved however as more useful.


possible translation-invariant mass in math \ the R^n /math classifies results <>the sentence of< Hadwiger>: the Lebesgue measure is likewise a special case like the Euler characteristic. Connections result furthermore to the Minkowski functional ones and the transverse masses.

See also

table of mathematical symbols


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