Category:Measures

This category contains results about measures.

Definition 1

$\mu$ is called a measure on $\Sigma$ if and only if $\mu$ fulfils the following axioms:

$(1)$	$:$	$\ds \forall E \in \Sigma:$	$\ds \map \mu E $	$\ds \ge $	$\ds 0 $
$(2)$	$:$	$\ds \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O:$	$\ds \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} $	$\ds = $	$\ds \sum_{n \mathop = 1}^\infty \map \mu {S_n} $	that is, $\mu$ is a countably additive function
$(3)$	$:$	$\ds \exists E \in \Sigma:$	$\ds \map \mu E $	$\ds \in $	$\ds \R $	that is, there exists at least one $E \in \Sigma$ such that $\map \mu E$ is finite

Definition 2

$\mu$ is called a measure on $\Sigma$ if and only if $\mu$ fulfils the following axioms:

$(1')$	$:$	$\ds \forall E \in \Sigma:$	$\ds \map \mu E $	$\ds \ge $	$\ds 0 $
$(2')$	$:$	$\ds \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O:$	$\ds \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} $	$\ds = $	$\ds \sum_{n \mathop = 1}^\infty \map \mu {S_n} $	that is, $\mu$ is a countably additive function
$(3')$	$:$		$\ds \map \mu \O $	$\ds = $	$\ds 0 $

Definition 3

$\mu$ is called a measure on $\Sigma$ if and only if $\mu$ fulfils the following axioms:

$(1' ')$	$:$	$\ds \forall E \in \Sigma:$	$\ds \map \mu E $	$\ds \ge $	$\ds 0 $
$(2' ')$	$:$	$\ds \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O:$	$\ds \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} $	$\ds = $	$\ds \sum_{n \mathop = 1}^\infty \map \mu {S_n} $	that is, $\mu$ is a countably additive function
$(3' ')$	$:$	$\ds S_i, S_j \in \Sigma, S_i \cap S_j = \O:$	$\ds \map \mu {S_i \cup S_j} $	$\ds = $	$\ds \map \mu {S_i} + \map \mu {S_j} $

Subcategories

This category has the following 23 subcategories, out of 23 total.

Pages in category "Measures"

The following 25 pages are in this category, out of 25 total.

C

D

Dirac Measure is Measure

I

K

Kernel Transformation of Measure is Measure

L

M

P

R

Restriction of Measure to Trace Sigma-Algebra of Measurable Set is Measure

S

Series of Measures is Measure

U

Uniqueness of Measures

V

Variation of Signed Measure is Measure

\((1)\)	$:$	\(\ds \forall E \in \Sigma:\)	\(\ds \map \mu E \)	\(\ds \ge \)	\(\ds 0 \)
\((2)\)	$:$	\(\ds \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O:\)	\(\ds \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} \)	\(\ds = \)	\(\ds \sum_{n \mathop = 1}^\infty \map \mu {S_n} \)	that is, $\mu$ is a countably additive function
\((3)\)	$:$	\(\ds \exists E \in \Sigma:\)	\(\ds \map \mu E \)	\(\ds \in \)	\(\ds \R \)	that is, there exists at least one $E \in \Sigma$ such that $\map \mu E$ is finite

\((1')\)	$:$	\(\ds \forall E \in \Sigma:\)	\(\ds \map \mu E \)	\(\ds \ge \)	\(\ds 0 \)
\((2')\)	$:$	\(\ds \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O:\)	\(\ds \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} \)	\(\ds = \)	\(\ds \sum_{n \mathop = 1}^\infty \map \mu {S_n} \)	that is, $\mu$ is a countably additive function
\((3')\)	$:$		\(\ds \map \mu \O \)	\(\ds = \)	\(\ds 0 \)

\((1' ')\)	$:$	\(\ds \forall E \in \Sigma:\)	\(\ds \map \mu E \)	\(\ds \ge \)	\(\ds 0 \)
\((2' ')\)	$:$	\(\ds \forall \sequence {S_n}_{n \mathop \in \N} \subseteq \Sigma: \forall i, j \in \N: S_i \cap S_j = \O:\)	\(\ds \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} \)	\(\ds = \)	\(\ds \sum_{n \mathop = 1}^\infty \map \mu {S_n} \)	that is, $\mu$ is a countably additive function
\((3' ')\)	$:$	\(\ds S_i, S_j \in \Sigma, S_i \cap S_j = \O:\)	\(\ds \map \mu {S_i \cup S_j} \)	\(\ds = \)	\(\ds \map \mu {S_i} + \map \mu {S_j} \)