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 $\struct {X, \Sigma, \mu}$

if and only if 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: $\bar \mu \restriction_\Sigma = \mu$