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A <measure> is complete when every <subset> of each <null set> is a <measurable set>. Such a subset also has measure zero by monotonicity. The <completion of a measure> extends it to a complete measure by adjoining all subsets of measurable null sets to its <sigma-algebra>.


Complete measure

The completion of a measure space adjoins every <subset> of every measurable <null set> to its <sigma-algebra>. A statement that two events or σ-algebras agree modulo null sets becomes literal after passing to the corresponding measure-algebra completion.


Completion of a measure

{title2=$\operatorname{ess\,inf}u$}

The <essential infimum> of a <measurable function> is $\operatorname{ess\,inf}u=-\operatorname{ess\,sup}(-u)$. It is the largest lower bound that holds <almost everywhere>. Unlike a pointwise <infimum>, it is unchanged by altering a representative on a <null set>.


Essential infimum

{title2=$\mu(N)=0$}
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A <measurable set> $N$ is a null set for a <measure> $\mu$ when $\mu(N)=0$. A property holds <almost everywhere> when its exceptions lie in a null set. With a <complete measure>, every <subset> of a null set is measurable and is itself a null set.


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 Incoming links: Null set