View source for Additive function

An additive function is a [[Homomorphism|homomorphism]] that preserves the operation of addition in place on the structure in question.

In group theory (because there's only one operation) it is usually just called a "[[Group|group]] [[Homomorphism|homomorphism]]"

==Definition==
Here {{M|(X,+_X:X\times X\rightarrow Y)}} (which we'll denote {{M|X}} and {{M|+_X}})  denotes a space with an additive function assigned.

The same goes for {{M|(Y,+_Y:Y\times Y\rightarrow Y)}}.

A function is additive<ref>http://en.wikipedia.org/w/index.php?title=Additive_function&oldid=630245379</ref> if for {{M|a,b\in X}}

<math>f(a+_Xb)=f(a)+_Yf(b)</math>
===Warning about structure===
If the spaces X and Y have some sort of structure (example: [[Group]]) then some required properties follow, for example:

<math>x=x+0\implies f(x)=f(x+0)=f(x)+f(0)\implies f(x)=f(x)+0\implies f(0)=0</math> so one must be careful!
===On set functions===
A [[Set function|set function]], {{M|\mu}}, is called additive if<ref>Halmos - p30 - Measure Theory - Springer - Graduate Texts in Mathematics (18)</ref> whenever:
* {{M|A\in X}}
* {{M|B\in X}}
* {{M|1=A\cap B=\emptyset}}

We have:

<math>\mu(A\cup B)=\mu(A)+\mu(B)</math> for valued set functions (set functions that map to values)

An example would be a [[Measure|measure]]

==Variations==
===Finitely additive===
This follows by induction on the additive property above. It states that:
* <math>f(\sum^n_{i=1}A_i)=\sum^n_{i=1}f(A_i)</math> for additive functions
* <math>\mu(\bigcup^n_{i=1}A_i)=\sum^n_{i=1}\mu(A_i)</math> for valued set functions

===Countably additive===
This is a separate property, while given additivity we can get finite additivity we cannot get additivity, we cannot get countable additivity from just additivity.

* <math>f(\sum^\infty_{n=1}A_n)=\sum^\infty_{n=1}f(A_n)</math> for additive functions
* <math>\mu(\bigcup^\infty_{n=1}A_n)=\sum^\infty_{n=1}\mu(A_n)</math> for valued set functions

====Countable additivity can imply additivity====
If <math>f(0)=0</math> or <math>\mu(\emptyset)=0</math> then given a finite set <math>\{a_i\}_{i=1}^n</math> we can define an infinite set <math>\{b_n\}_{n=1}^\infty</math> by:

<math>b_i=\left\{\begin{array}a_i&\text{if }i\le n\\ 0\text{ or }\emptyset & \text{otherwise}\end{array}\right.</math>

Thus:
* <math>f(\sum^\infty_{n=1}b_n)=
\begin{array}{lr}
f(\sum^n_{i=1}a_i) \\ 
\sum^\infty_{n=1}f(b_n)=\sum^n_{i=1}f(a_i)+f(0)=\sum^n_{i=1}f(a_i)
\end{array}</math>
* Or indeed <math>\mu(\sum^\infty_{n=1}b_n)=
\begin{array}{lr}
\mu(\sum^n_{i=1}a_i) \\ 
\sum^\infty_{n=1}\mu(b_n)=\sum^n_{i=1}\mu(a_i)+\mu(0)=\sum^n_{i=1}\mu(a_i)
\end{array}</math>

{{Definition|Abstract Algebra|Measure Theory}}

==References==