Additive function

In mathematics, functions are called additive when they receive sums, ie

Are definition and target area abelian groups, so one speaks also of linearity.

Sub - and superadditivity

If a semigroup with the operation " " is the name of a figure subadditive if and only if x and y from for all:

The mapping is called superadditive if and out applies to all:

A picture is exactly then additive when it is both sub - and superadditive.

Examples

  • According to the triangle inequality standards and amounts are always subadditive.
  • Sublinear functions are subadditive.
  • Linear mappings additive.

Properties

If an additive function, then for any finite number of elements from:

The same goes for sub - and super-additivity.

Definition in number theory

In number-theoretic functions is considered as a shortcut to the multiplication. A number-theoretic function is called additive when the equation

For all prime and apply. Does this apply even for all, and so the function be strictly additive.

A similar restriction of additivity (on disjoint take arbitrary unions ) are available in measure theory, see next section.

Sigma - additivity, dimensions

Is a σ - algebra and thus in particular a semigroup with respect to the link ( association ), it means a function countably additive or σ - additive when a corresponding equation infinite number of disjoint sets is also true for countable, ie:

General is an outer measure σ - subadditive, that is, it is:

For all countable families.

An inner measure, however, is σ -super additive for disjoint unions, that is, it is:

For all countable family of disjoint sets in.

A picture that is both an outer and an inner measure, is a measure. Dimensions are therefore σ - additive for disjoint unions, for arbitrary unions but only σ - subadditive.

  • Measure theory
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