View source for Additive function

{{Refactor notice}}
{{Requires references|See Halmos' measure theory book too}}
{{Stub page|Needs to include everything the old page did, link to propositions and lead to measures}}
==Definition==
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-->{{Extra Maths}}A ''[[real valued function|real valued]]'' [[set function]] on a class of [[set|sets]], {{M|\mathcal{A} }}, {{M|f:\mathcal{A}\rightarrow\mathbb{R} }} is called ''additive'' or ''finitely additive'' if{{rMT1VIB}}:
* For {{M|A,B\in\mathcal{A} }} with {{M|1=A\cap B=\emptyset}} ([[pairwise disjoint]]) and {{M|A\udot B\in\mathcal{A} }} we have:
** {{M|1=f(A\udot B)=f(A)+f(B)}}
===Finitely additive===
With the same definition of {{M|f}}, we say that {{M|f}} is ''finitely additive'' if for a [[pairwise disjoint]] family of sets {{M|1=\{A_i\}_{i=1}^n\subseteq\mathcal{A} }} with {{M|1=\bigudot_{i=1}^nA_i\in\mathcal{A} }} we have<ref name="MT1VIB"/>:
* {{MM|1=f\left(\mathop{\bigudot}_{i=1}^nA_i\right)=\sum^n_{i=1}f(A_i)}}.
'''Claim 1: ''' {{M|f}} is finitely additive {{M|\iff}} it is additive
===Countably additive===
With the same definition of {{M|f}}, we say that {{M|f}} is ''countably additive'' if for a [[pairwise disjoint]] family of sets {{M|1=\{A_n\}_{n=1}^\infty\subseteq\mathcal{A} }} with {{M|1=\bigudot_{n=1}^\infty A_n\in\mathcal{A} }} we have<ref name="MT1VIB"/>:
* {{MM|1=f\left(\mathop{\bigudot}_{n=1}^\infty A_n\right)=\sum^\infty_{n=1}f(A_n)}}.
==Immediate properties==
{{Begin Inline Theorem}}
'''Claim: ''' if {{M|\emptyset\in\mathcal{A} }} then {{M|1=f(\emptyset)=0}}
{{Begin Inline Proof}}
Let {{M|\emptyset,A\in\mathcal{A} }}, then:
* {{M|1=f(A)=f(A\udot\emptyset)=f(A)+f(\emptyset)}} by hypothesis. 
* Thus {{M|1=f(A)=f(A)+f(\emptyset)}]
* This means {{M|1=f(A)-f(A)=f(\emptyset)}}
We see {{M|1=f(\emptyset)=0}}, as required
{{End Proof}}{{End Theorem}}
==Proof of claims==
{{Begin Inline Theorem}}
'''[[Additive function/Claim 1 - additive iff finitely additive|Claim 1]]: ''' {{M|f}} is ''additive'' {{iff}} {{M|f}} is ''finitely additive''
{{Begin Inline Proof}}
{{:Additive function/Claim 1 - additive iff finitely additive}}
{{End Proof}}{{End Theorem}}
==See also==
* [[Subadditive function]]
==Notes==
<references group="Note"/>
==References==
<references/>
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{{Definition|Measure Theory}}


{{Todo|Check algebra books for definition of additive, perhaps split into two cases, additive set function and additive function}}

=OLD PAGE=
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 X)}} (which we'll denote {{M|X}} and {{M|+_X}})  denotes a set endowed with a binary operation called addition.

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

A function {{M|f}} 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)+0=f(x)=f(x+0)=f(x)+f(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)

A shorter notation:
<math>\mu(A\uplus B)=\mu(A)+\mu(B)</math>, where {{M|\uplus}} denotes "disjoint union" -- just the union when the sets are disjoint, otherwise undefined.

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

==Variations==
===Finitely additive===
This follows by induction on the additive property above. It states that:
* <math>f\Big(\sum^n_{i=1}A_i\Big)=\sum^n_{i=1}f(A_i)</math> for additive functions
* <math>\mu\Big(\biguplus^n_{i=1}A_i\Big)=\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, but we cannot get countable additivity from just additivity.

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