Carathéodory's Theorem (Measure Theory)
This article is a landmark page. It was the 5000th proof on ProofWiki!This proof is about Carathéodory's Theorem in measure theory. For other uses, see Carathéodory's Theorem.
Contents |
Theorem
Let $X$ be a set, and let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a semi-ring of subsets of $X$.
Let $\mu: \mathcal S \to \overline{\R}$ be a pre-measure on $\mathcal S$.
Let $\sigma \left({\mathcal S}\right)$ be the $\sigma$-algebra generated by $\mathcal S$.
Then $\mu$ extends to a measure $\mu^*$ on $\sigma \left({\mathcal S}\right)$.
Corollary
Suppose there exists an exhausting sequence $\left({S_n}\right)_{n \in \N} \uparrow X$ in $\mathcal S$ such that:
- $\forall n \in \N: \mu \left({S_n}\right) < +\infty$
Then the extension $\mu^*$ is unique.
Proof
You can help Proof Wiki by expanding it.
To discuss it in more detail, feel free to use the talk page.
When this page/section has been completed, {{Stub}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Source of Name
This entry was named for Constantin Carathéodory.
