Definition:Completion (Measure Space)

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Definition

Let $\left({X, \Sigma, \mu}\right), \left({\tilde X, \Sigma^*, \bar \mu}\right)$ be measure spaces.

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

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


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