Essential infimum and essential supremum

In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for all elements in a set, but rather almost everywhere, that is, except on a set of measure zero.

While the exact definition is not immediately straightforward, intuitively the essential supremum of a function is the smallest value that is greater than or equal to the function values everywhere while ignoring what the function does at a set of points of measure zero. For example, if one takes the function ${\displaystyle f(x)}$ that is equal to zero everywhere except at ${\displaystyle x=0}$ where ${\displaystyle f(0)=1,}$ then the supremum of the function equals one. However, its essential supremum is zero if we apply the Lebesgue-Borel measure and are allowed to ignore what the function does at the single point where ${\displaystyle f}$ is peculiar. The essential infimum is defined in a similar way.