Definition:Measurable Set/Arbitrary Outer Measure
Jump to navigation
Jump to search
Definition
Let $\struct {X, \Sigma}$ be a measurable space.
![]() | Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: $\Sigma$ is not used. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Let $\mu^*$ be an outer measure on $X$.
A subset $S \subseteq X$ is called $\mu^*$-measurable if and only if it satisfies the Carathéodory condition:
- $\map {\mu^*} A = \map {\mu^*} {A \cap S} + \map {\mu^*} {A \setminus S}$
for every $A \subseteq X$.
![]() | This definition needs to be completed. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{DefinitionWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
The set of $\mu^*$-measurable sets is denoted $\map {\mathfrak M} {\mu^*}$ and is a $\sigma$-algebra over $X$.
Sources
![]() | This page may be the result of a refactoring operation. As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering. In particular: Refactored page now has 3 subpages If you have access to any of these works, then you are invited to review this list, and make any necessary corrections. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{SourceReview}} from the code. |
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $3.1$