Revision as of 08:13, 19 March 2016 (view source) Boris (Talk | contribs) m (→‎Countably additive: typo, notation, format) ← Older edit		Revision as of 22:34, 19 March 2016 (view source) Alec (Talk | contribs) (Refactored page and cleaned up into standard definition form. Definition is more concrete.) Newer edit →
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		+	{{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==
		+	{{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/>
		+	{{Measure theory navbox|plain}}
		+	{{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.		An additive function is a [[Homomorphism|homomorphism]] that preserves the operation of addition in place on the structure in question.

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Revision as of 22:34, 19 March 2016

Definition

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$ A real valued set function on a class of sets, $\mathcal{A}$, $f:\mathcal{A}\rightarrow\mathbb{R}$ is called additive or finitely additive if^[1]:

• For $A,B\in\mathcal{A}$ with $A\cap B=\emptyset$ (pairwise disjoint) and $A\udot B\in\mathcal{A}$ we have:
  • $f(A\udot B)=f(A)+f(B)$

Finitely additive

With the same definition of $f$, we say that $f$ is finitely additive if for a pairwise disjoint family of sets $\{A_i\}_{i=1}^n\subseteq\mathcal{A}$ with $\bigudot_{i=1}^nA_i\in\mathcal{A}$ we have^[1]:

• $$f\left(\mathop{\bigudot}_{i=1}^nA_i\right)=\sum^n_{i=1}f(A_i)$$ .

Claim 1: $f$ is finitely additive $\iff$ it is additive

Countably additive

With the same definition of $f$, we say that $f$ is countably additive if for a pairwise disjoint family of sets $\{A_n\}_{n=1}^\infty\subseteq\mathcal{A}$ with $\bigudot_{n=1}^\infty A_n\in\mathcal{A}$ we have^[1]:

• $$f\left(\mathop{\bigudot}_{n=1}^\infty A_n\right)=\sum^\infty_{n=1}f(A_n)$$ .

Immediate properties

Proof of claims

See also

• Subadditive function

Notes

References

1. ↑ ^{Jump up to: 1.0 1.1 1.2} Measure Theory - Volume 1 - V. I. Bogachev

v • d • e   Measure Theory Overview of the basic and important objects studied in measure theory Set-structures ring of sets, algebra of sets, $\sigma$-ring, $\sigma$-algebra, hereditary $\sigma$-ring measurable space Measures Pre-measure $\leadsto$ outer-measure $\leadsto$ measure (see Extending pre-measures to measures (via: Extending pre-measures to outer-measures $\rightarrow$ "Restricting outer-measures to measures" maybe?). Alternatively: inner-measure can be used. measure space Functions measurable map, simple function (standard representation) Spaces Integrals Integrating a simple function $\leadsto$ Integral of a positive function $\leadsto$ Integrating all measurable functions

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TODO: Check algebra books for definition of additive, perhaps split into two cases, additive set function and additive function

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OLD PAGE

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"

Definition

Here $(X,+_X:X\times X\rightarrow X)$ (which we'll denote $X$ and $+_X$) denotes a set endowed with a binary operation called addition.

The same goes for $(Y,+_Y:Y\times Y\rightarrow Y)$.

A function $f$ is additive^[1] if for $a,b\in X$

$$f(a+_Xb)=f(a)+_Yf(b)$$

Warning about structure

If the spaces X and Y have some sort of structure (example: Group) then some required properties follow, for example:

$$x=x+0\implies f(x)+0=f(x)=f(x+0)=f(x)+f(0)\implies f(0)=0$$ so one must be careful!

On set functions

A set function, $\mu$, is called additive if^[2] whenever:

• $A\in X$
• $B\in X$
• $A\cap B=\emptyset$

We have:

$$\mu(A\cup B)=\mu(A)+\mu(B)$$ for valued set functions (set functions that map to values)

A shorter notation:

$$\mu(A\uplus B)=\mu(A)+\mu(B)$$ , where $\uplus$ denotes "disjoint union" -- just the union when the sets are disjoint, otherwise undefined.

An example would be a measure.

Variations

Finitely additive

This follows by induction on the additive property above. It states that:

• $$f\Big(\sum^n_{i=1}A_i\Big)=\sum^n_{i=1}f(A_i)$$ for additive functions
• $$\mu\Big(\biguplus^n_{i=1}A_i\Big)=\sum^n_{i=1}\mu(A_i)$$ 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.

• $$f\Big(\sum^\infty_{n=1}A_n\Big)=\sum^\infty_{n=1}f(A_n)$$ for additive functions
• $$\mu\Big(\biguplus^\infty_{n=1}A_n\Big)=\sum^\infty_{n=1}\mu(A_n)$$ for valued set functions

Countable additivity can imply additivity

If

$$f(0)=0$$ or $$\mu(\emptyset)=0$$ then given a finite set $$\{a_i\}_{i=1}^n$$ we can define an infinite set $$\{b_n\}_{n=1}^\infty$$ by:

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

Thus:

• $$f(\sum^\infty_{n=1}b_n)= \begin{array}{lr} f(\sum^n_{i=1}a_i) \\ \sum^\infty_{n=1}f(b_n)=\sum^n_{i=1}f(a_i)+f(0)=\sum^n_{i=1}f(a_i) \end{array}$$
• Or indeed $$\mu(\sum^\infty_{n=1}b_n)= \begin{array}{lr} \mu(\sum^n_{i=1}a_i) \\ \sum^\infty_{n=1}\mu(b_n)=\sum^n_{i=1}\mu(a_i)+\mu(0)=\sum^n_{i=1}\mu(a_i) \end{array}$$

References

1. Jump up ↑ http://en.wikipedia.org/w/index.php?title=Additive_function&oldid=630245379
2. Jump up ↑ Halmos - p30 - Measure Theory - Springer - Graduate Texts in Mathematics (18)