Definition:Completion (Measure Space)

Definition
Let $\struct {X, \Sigma, \mu}, \struct {\tilde X, \Sigma^*, \bar \mu}$ be measure spaces.

Then:
 * $\struct {\tilde X, \Sigma^*, \bar \mu}$ is a completion of $\struct {X, \Sigma, \mu}$

or:
 * $\struct {\tilde X, \Sigma^*, \bar \mu}$ completes $\struct {X, \Sigma, \mu}$

the following conditions hold:


 * $(1): \quad \struct {\tilde X, \Sigma^*, \bar \mu}$ is a complete measure space
 * $(2): \quad \tilde X = X$
 * $(3): \quad \Sigma$ is a sub-$\sigma$-algebra of $\Sigma^*$
 * $(4): \quad \forall E \in \Sigma: \map {\bar \mu} E = \map \mu E$, that is: $\bar \mu \restriction_\Sigma = \mu$

Also see

 * Completion Theorem (Measure Space), demonstrating that any measure space can be completed