Sup VS esssup, inf VS essinf

Supremum

In mathematics, given a subset S of a partially ordered set T, the supremum (sup) of S, if it exists, is the least element of T that is greater than or equal to each element of S. Consequently, the supremum is also referred to as the least upper bound, lub or LUB. If the supremum exists, it may or may not belong to S. If the supremum exists, it is unique.

http://upload.wikimedia.org/wikipedia/commons/thumb/0/03/Supremum_illustration.png/300px-Supremum_illustration.png

Infimum

In mathematics, particularly set theory, the infimum (plural infima) of a subset of some set is the greatest element (not necessarily in the subset) that is less than or equal to all elements of the subset. Consequently the term greatest lower bound (also abbreviated as glb or GLB) is also commonly used. Infima of real numbers are a common special case that is especially important in analysis. However, the general definition remains valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.

Infima are in a precise sense dual to the concept of a supremum.
http://upload.wikimedia.org/wikipedia/en/thumb/0/0a/Infimum_illustration.svg/300px-Infimum_illustration.svg.png

Essential supremum and essential infimum
\mathrm{ess } \sup f=\inf \{a \in \mathbb{R}: \mu(\{x: f(x) > a\}) = 0\}\,
\mathrm{ess } \inf f=\sup \{b \in \mathbb{R}: \mu(\{x: f(x) < b\}) = 0\}\,