Carathéodory’s lemma
In measure theory, Carathéodory’s lemma is used for constructing measures^{} and, for example, can be applied to the construction of the Lebesgue measure^{} and is used in the proof of Carathéodory’s extension theorem. The idea is that to define a measure on a measurable space^{} $(X,\mathcal{A})$ we would first construct an outer measure^{} (http://planetmath.org/OuterMeasure2), which is a set function^{} defined on the power set^{} of $X$. Then, this outer measure is restricted to $\mathcal{A}$ and Carathéodory’s lemma is applied to show that this restriction^{} does in fact result in a measure. For an example of this procedure, see the proof of Carathéodory’s extension theorem.
Given an outer measure $\mu $ on a set $X$, the result first defines a collection^{} of subsets of $X$ — the $\mu $-measurable sets. A subset $S\subseteq X$ is called $\mu $-measurable (or Carathéodory measurable with respect to $\mu $) if the equality
$$\mu (E)=\mu (E\cap S)+\mu (E\cap {S}^{c})$$ |
holds for every $E\subseteq X$. Then, Caratheodory’s lemma says that a measure is obtained by restricting $\mu $ to the $\mu $-measurable sets.
Lemma (Carathéodory).
Let $\mu $ be an outer measure on a set $X$, and $\mathrm{A}$ be the class of $\mu $-measurable sets. Then $\mathrm{A}$ is a $\sigma $-algebra (http://planetmath.org/SigmaAlgebra) and the restriction of $\mu $ to $\mathrm{A}$ is a measure.
It should be noted that for any outer measure $\mu $ and sets $S,E\subseteq X$, subadditivity of $\mu $ implies that the inequality^{} $\mu (E)\le \mu (E\cap S)+\mu (E\cap {S}^{c})$ is always satisfied. So, only the reverse inequality is required and consequently $S$ is $\mu $-measurable if and only if
$$\mu (E)\ge \mu (E\cap S)+\mu (E\cap {S}^{c})$$ |
for every $E\subseteq X$.
References
- 1 David Williams, Probability with martingales^{}, Cambridge Mathematical Textbooks, Cambridge University Press, 1991.
- 2 Olav Kallenberg, Foundations of modern probability, Second edition. Probability and its Applications. Springer-Verlag, 2002.
Title | Carathéodory’s lemma |
Canonical name | CaratheodorysLemma |
Date of creation | 2013-03-22 18:33:03 |
Last modified on | 2013-03-22 18:33:03 |
Owner | gel (22282) |
Last modified by | gel (22282) |
Numerical id | 19 |
Author | gel (22282) |
Entry type | Theorem |
Classification | msc 28A12 |
Related topic | CaratheodorysExtensionTheorem |
Related topic | OuterMeasure2 |
Related topic | LebesgueOuterMeasure |
Related topic | ConstructionOfOuterMeasures |
Related topic | ProofOfCaratheodorysLemma |
Related topic | ProofOfCaratheodorysExtensionTheorem |
Defines | Carathéodory measurable |