Which means almost everywhere in mathematics

Measure theory

The Measure theory is a branch of mathematics, which generalizes the elementary geometric terms length, area, volume and thus also makes it possible to enter more complicated sets Measure assign. It forms the foundation of modern integration and probability theory.
In measure theory, a measure is an assignment of real or complex numbers to a subset system over a basic set. The assignment and the subset system should have certain properties. In practice, often only a partial assignment is known in advance. For example, in the Rectangles plane, you assign the product of their edge lengths as the area. 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, translation invariance is also required, i.e. the content of a subset of the plane is independent of its position.

Definitions and examples

Measuring room, measurable quantities

For an exact definition of the basic concepts of measure theory, we start with a basic set Ω. If a certain setΣ of subsets of Ω forms a σ-algebra, then every set that is an element of Σ is called measurable (engl. measurable), and the basic set is called Ω with the structure Σ Measuring room (engl. measurable space). A function that maintains the structure of a measuring room is called a measurable function.
The requirement that Σ a σ-algebra is, means
  • that jeder with every set S also contains its complement Ω \ S,
  • that Σ contains the empty set (and thus also its complement Ω), and
  • that Σ is closed with respect to the countable union.
  • Every finite or countable infinite set, especially the set of natural numbers N, forms a measurement space with its power set as a σ-algebra.
  • If A is a subset of Ω, then {Ω, ∅, A, Ω ∖ A} is a σ-algebra.

Measure, measure space

A Measureμ is a function that assigns a value μ (S) to every set S in Σ. This value is either a nonnegative real number or ∞ (see below for possible generalizations). The following must also apply:
  • The empty set has the measure zero: μ (∅) = 0.
  • The measure is countable additive (also σ-additive), that is, if E1, E2, E3, ... are countably many pairwise disjoint sets from Σ and E is their union, then the measureμ (E) is equal to the sum ∑μ (Ek).
The structure (Ω, Σ, μ) of a measurement space together with a dimension defined on this is called a measurement space. measure space).

Zero quantity, complete, almost everywhere

A Zero quantity is a set S from Σ with the measure μ (S) = 0. A measure is called Completelyif every subset of every null set is contained in Σ. One property applies almost everywhere in Ω, if it is only not valid in a null set.
Examples of zero quantities:

finite, σ-finite

A measure is called at lastif μ (Ω) <∞. A measure is called σ-finite, if Ω is the union of a countable sequential measurable set {S1, S2, S3,…}, all of which have a finite measureμ (Sk) <∞.
σ-finite measures have some nice properties that have certain analogies to the properties of separable topological spaces.

Examples

  • The zero measure that assigns the value µ (S) = 0 to each set S.
  • The counting measure assigns the number of its elements to each subset S of a finite or countable infinite set, µ (S) = | S |.
  • The Lebesgue measure on the set of real numbers R with Borel's σ-algebra, defined as a translation-invariant measure with μ ([0,1]) = 1.
  • The hair measure on locally compact topological groups.
  • Probability measures, with μ (Ω) = 1.
  • The counting measure on the set N of natural numbers is infinite, but σ-finite.
  • The canonical Lebesgue measure on the set R of real numbers is also infinite, but σ-finite, because R can be represented as the union of countable many finite intervals [k, k + 1].
  • The Lévy measure is a random measure(random measure) and is needed, among other things, to characterize Lévy processes. It gives the expected number of jumps of the process of this height in the unit interval.

Generalizations

  • Negative real or complex values ​​can be admitted (complex or signed dimension).
  • Another example of a generalization is the spectral measure, the values ​​of which are linear operators. This measure is used especially in functional analysis for the spectral theorem.
Another possibility of generalization is the definition of a measure on the power set.
Historically they were first finally additiveDimensions introduced. The modern definition, hence a measurecountable additive is, however, turned out to be more useful.

Results

The Hadwiger's theorem classifies all possible translation-invariant measures in Rn: the Lebesgue measure is just as special as the Euler characteristic. There are also connections to the Minkowski functionals and the cross-measures.

See also

"Obviously" is the most dangerous word in math.

Eric Temple Bell

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