Measures: Difference between revisions
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Let <math>X</math> be a set equipped with a <math>\sigma</math>-algebra <math>\mathcal{M}</math>. A '''measure on <math>(X, \mathcal{M})</math>''' (also referred to simply as '''measure on <math>X</math>''' if <math>M</math> is understood) is a function <math>\mu: \mathcal{M} \rightarrow [0, \infty]</math> that satisfies the following criteria: | Let <math>X</math> be a set equipped with a <math>\sigma</math>-algebra <math>\mathcal{M}</math>. A '''measure on <math>(X, \mathcal{M})</math>''' (also referred to simply as '''measure on <math>X</math>''' if <math>M</math> is understood) is a function <math>\mu: \mathcal{M} \rightarrow [0, \infty]</math> that satisfies the following criteria: | ||
# <math>\mu\left(\emptyset\right) = 0</math>, | # <math>\mu\left(\emptyset\right) = 0</math>, | ||
# Let <math>\left\{E_k\right\}_{k = 1}^{\infty}</math> be a disjoint sequence of sets such that <math>E_k \in \mathcal{M}</math>. Then, <math>\mu\left(\cup_{k = 1}^{\infty} E_k\right) = \sum_{k = 1}^{\infty} \mu\left(E_k\right)</math>. | # Let <math>\left\{E_k\right\}_{k = 1}^{\infty}</math> be a disjoint sequence of sets such that each <math>E_k \in \mathcal{M}</math>. Then, <math>\mu\left(\cup_{k = 1}^{\infty} E_k\right) = \sum_{k = 1}^{\infty} \mu\left(E_k\right)</math>. | ||
==Properties== | ==Properties== | ||
Revision as of 16:34, 16 December 2020
This page is under construction.
Definition
Let be a set equipped with a -algebra . A measure on (also referred to simply as measure on if is understood) is a function that satisfies the following criteria:
![{\displaystyle \mu :{\mathcal {M}}\rightarrow [0,\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c67f1b7f775328f952afd48b62cc8d2af2f028)
- ,
- Let be a disjoint sequence of sets such that each . Then, .
Properties
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Examples
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References
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