Set function

In mathematics, quantity functions are functions that assign certain amounts ( the amounts of a lot of system ) values ​​, usually non-negative real numbers or the value.

Lot of features form the basis for the measure theory, where examined, among other set functions, such as dimensions or content on detailed properties.

Motivation

Lot of features are particularly important in measure theory. Idea of measure theory is to assign quantities a (real) measure. A simple example would be to count the elements of a finite set: The set about then receives degree 4

Now you can explore lot of features on their properties. In measure theory one demands frequent certain stability properties, such as the additivity, that is, that if you cut a lot, so the two new quantities must adopt together the same value as the initial amount. This is fulfilled in the above example, when counting, it is.

Formal definition

Let be a non-empty set and a lot of system. Next was initially short. Then we call each picture with a lot of function.

Given a set function is called in most cases also, if or ( signed measure ) or ( complex measure ).

Examples

  • Certain sets of points of the plane ( the surface ) can be assigned as a measure an area. This assignment is (as well as the previous one) is always greater or equal to 0 and σ - additive; so a lot of function is called a measure.
  • In the analysis, the area between the x-axis and a function graph with the help of the integral is determined. It faces get below the x - axis has a negative sign. This assignment is σ - additive; so a lot of function is called a signed measure.
  • Probability measures are σ - additive set functions that take values ​​between 0 and 1, and the total base quantity level 1 assign ( " certain event ").
  • An Outer measure is a σ - subadditive set function, which is always greater than or equal to 0. This is achieved for example by assigning each subset of the plane, the infimum of the areas of all surfaces as measurable supersets. Mostly, however, it 's going the other way around and constructed an outer measure to obtain by suitable restriction of the measurable quantities of a measure (eg, construction of the Lebesgue measure ).

Special properties of set functions

The set function f is called:

Main Features

Compatibility of addition and union

Continuity

Applies.

Applies.

Applies.

Relationships between the properties

  • Each σ - additive set function is finite additive and any finite additive set function is additive.
  • Every finite set function is σ - finite.
  • Each additive set function is subtractive.
  • Every bounded set function is finite.
  • Is a ring so each additive set function is finite additive and any sub-additive set function is finally subadditive.
Disjoint sets
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