Definition:Completion (Measure Space)

Definition
Let $\left({X, \mathcal A, \mu}\right), \left({\tilde X, \mathcal A^*, \tilde \mu}\right)$ be measure spaces.

Then $\left({\tilde X, \mathcal A^*, \tilde \mu}\right)$ is a completion of $\left({X, \mathcal A, \mu}\right)$ or $\left({\tilde X, \mathcal A^*, \tilde \mu}\right)$ completes $\left({X, \mathcal A, \mu}\right)$ iff the following conditions hold:


 * $(1):\quad \left({\tilde X, \mathcal A^*, \tilde \mu}\right)$ is a complete measure space
 * $(2):\quad X = \tilde X$
 * $(3):\quad \mathcal A$ is a sub-$\sigma$-algebra of $\mathcal A^*$
 * $(4):\quad \forall A \in \mathcal A: \tilde \mu \left({A}\right) = \mu \left({A}\right)$, i.e. $\tilde \mu \restriction_{\mathcal A} = \mu$

Also see

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