Carathéodory's Theorem (Measure Theory)/Corollary
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Corollary to Carathéodory's Theorem (Measure Theory)
Let $X$ be a set.
Let $\SS \subseteq \powerset X$ be a semi-ring of subsets of $X$.
Let $\mu: \SS \to \overline \R$ be a pre-measure on $\SS$.
Let $\map \sigma \SS$ be the $\sigma$-algebra generated by $\SS$.
Suppose there exists an exhausting sequence $\sequence {S_n}_{n \mathop \in \N} \uparrow X$ in $\SS$ such that:
- $\forall n \in \N: \map \mu {S_n} < +\infty$
Then the extension $\mu^*$ is unique.
Proof
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Source of Name
This entry was named for Constantin Carathéodory.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $6.1$