Difference between revisions of "Additive function"
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Thus: | Thus: | ||
* <math>f(\sum^\infty_{n=1}b_n)= | * <math>f(\sum^\infty_{n=1}b_n)= | ||
| − | \begin{array} | + | \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) | \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> | \end{array}</math> | ||
* Or indeed <math>\mu(\sum^\infty_{n=1}b_n)= | * Or indeed <math>\mu(\sum^\infty_{n=1}b_n)= | ||
| − | \begin{array} | + | \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) | \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> | \end{array}</math> | ||
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{{Definition|Abstract Algebra|Measure Theory}} | {{Definition|Abstract Algebra|Measure Theory}} | ||
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==References== | ==References== | ||
Revision as of 16:54, 13 March 2015
An additive function is a 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 homomorphism"
Contents
Definition
Here [ilmath](X,+_X:X\times X\rightarrow Y)[/ilmath] (which we'll denote [ilmath]X[/ilmath] and [ilmath]+_X[/ilmath]) denotes a space with an additive function assigned.
The same goes for [ilmath](Y,+_Y:Y\times Y\rightarrow Y)[/ilmath].
A function is additive[1] if for [ilmath]a,b\in X[/ilmath]
[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, [ilmath]\mu[/ilmath], is called additive if[2] whenever:
- [ilmath]A\in X[/ilmath]
- [ilmath]B\in X[/ilmath]
- [ilmath]A\cap B=\emptyset[/ilmath]
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
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]
References
- ↑ http://en.wikipedia.org/w/index.php?title=Additive_function&oldid=630245379
- ↑ Halmos - p30 - Measure Theory - Springer - Graduate Texts in Mathematics (18)
