 Essential supremum and essential infimum

In mathematics, the concepts of essential supremum and essential infimum are related to the notions of supremum and infimum, 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", i.e., 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 larger or equal than the function values everywhere when allowing for ignoring what the function does at a set of points of measure zero.For example, if one takes the function formula_1 that is equal to zero everywhere except at formula_2 where formula_3, then the supremum of the function equals one.