Definition:Complete Measure Space
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Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let the family of $\mu$-null sets $\NN_\mu$ satisfy the condition:
- $\forall N \in \NN_\mu: \forall M \subseteq N: M \in \NN_\mu$
That is, any subset of a $\mu$-null set is again a $\mu$-null set.
Then $\struct {X, \Sigma, \mu}$ is said to be a complete measure space.
Also see
- Completion Theorem (Measure Space), showing that any measure space may be embedded in a complete measure space
- Results about complete measure spaces can be found here.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 4$: Problem $13$
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $1.5$: Completeness and Regularity