Knowlpedia

Definition

Completion of a measure space

The extension that makes all subsets of existing null sets measurable.

The completion of (X,Σ,μ)(X,\Sigma,\mu) uses the sigma-algebra

Σ‾={E⊆X:A⊆E⊆B for some A,B∈Σ with μ(B∖A)=0}.\overline\Sigma=\{E\subseteq X:A\subseteq E\subseteq B \text{ for some }A,B\in\Sigma\text{ with }\mu(B\setminus A)=0\}.

Set μ‾(E)=μ(A)\overline\mu(E)=\mu(A) for any such pair. This is well-defined because any two lower approximations differ only by null sets, and μ‾\overline\mu extends μ\mu. It makes the space .

Lebesgue measure

Completing Borel Lebesgue measure adds all subsets of Borel null sets. Likewise, completing the product of one-dimensional Lebesgue measures gives full Lebesgue measure in the product Euclidean space. Completion changes the measurable sets but not the values on sets measurable before the extension.