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

= Essential infima
{synonym}

The <essential infimum> of a real <measurable function> is $\operatorname{ess\,inf}_\mu f=-\operatorname{ess\,sup}_\mu(-f)$. It is the largest lower bound that holds <almost everywhere>, with values allowed in the <extended real number line>. Unlike a pointwise <infimum>, it is unchanged by altering a representative on a <null set>.

For the <function> $g$ in <essential infimum and essential supremum>, the value $-3$ at one input changes the pointwise <infimum>, but the <essential infimum> remains zero under <Lebesgue measure>. If instead a <function> is $-3$ throughout an <interval> of any positive length and zero elsewhere, its <essential infimum> is $-3$. The distinction concerns the <measure> of the low-value <set>, rather than the magnitude of the low value.


 Essential infimum (source code)