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