Complete measure - Knowlpedia

A measure space $(X,\Sigma,\mu)$ is complete if every subset of every measurable set of measure zero belongs to $\Sigma$ . Such subsets necessarily have measure zero by monotonicity.

Two notions of completeness

This property concerns the sigma-algebra and its null sets. It is distinct from completeness of a metric or Banach space. An incomplete measure space still has complete $L^p$ spaces: metric completeness of $L^p$ does not require the measure to contain all subsets of null sets.