# 本性上界与 L^∞　`def.essential-sup`
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> English: **draft** · machine-drafted, pending author review; the Chinese page is authoritative.

layer 16 · definition · cluster L^p 空间 (`analysis.lp`) · Analysis

## Abstract
The essential supremum is $\|f\|_\infty := \inf \{a \ge 0 : \mu(\{|f| > a\}) = 0\}$: the true bound of $f$ once its values on null sets are ignored. The infimum is attained at $a = \|f\|_\infty$ itself, and $\|f\|_\infty \le M \iff |f| \le M$ almost everywhere. Then $L^\infty$ consists of the measurable $f$ with $\|f\|_\infty < \infty$. Note the difference from $\sup |f|$, which a single value on a null set can spoil.

> 续见 nodes/def.essential-sup.en.2.md