\begin{definition}[Definition:Completion (Measure Space)]
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 $\struct {X, \Sigma, \mu}$'''
{{iff}} 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: $\bar \mu \restriction_\Sigma = \mu$
\end{definition}