Category:Definitions/Measures
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This category contains definitions related to Measures.
Related results can be found in Category: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 4 subcategories, out of 4 total.
Pages in category "Definitions/Measures"
The following 21 pages are in this category, out of 21 total.