concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis...

the supremum is also referred to as the least upper bound (or LUB). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema...

valued function equals the supremum of its essential image and the essential infimum equals the infimum of its essential range. Consequently, a function...

for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are...

limit infimum and limit supremum of a set sequence always exist and can be used to determine convergence: the limit exists if the limit infimum and limit...

of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural...

{\mathbb {R} }}} has a supremum and an infimum (the infimum of the empty set is + ∞ {\displaystyle +\infty } , and its supremum is − ∞ {\displaystyle -\infty...

bound. Essential supremum and essential infimum Initial and terminal objects Maximal and minimal elements Limit superior and limit inferior (infimum limit)...

then the supremum and infimum are equal to each other and a unique hedging price exists. The upper and lower bounds created by the subhedging and superhedging...

I {\displaystyle I} in its domain is the difference between the supremum and infimum of f {\displaystyle f} : ω f ( I ) = sup x ∈ I f ( x ) − inf x ∈...

their composition non-Lebesgue-measurable. The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) of real-valued...

, {\displaystyle 0,} substituting y := 0 {\displaystyle y:=0} in this infimum shows that d ( u , Y ) ≤ ‖ u ‖ {\displaystyle d(u,Y)\leq \|u\|} for every...

between any two points x and y is the infimum of lengths of paths between them. Unlike in a geodesic metric space, the infimum does not have to be attained...

) ≠ 0 {\displaystyle \mu (S)\neq 0} then this is the same as the essential supremum of the absolute value of f {\displaystyle f} : ‖ f ‖ ∞   =   { esssup...

{\displaystyle \gamma \operatorname {-essup} d(x,y)} denotes the essential supremum of d ( x , y ) {\displaystyle d(x,y)} with respect to measure γ {\displaystyle...

{\displaystyle x} then the discontinuity will be a removable discontinuity, or an essential discontinuity, or a jump discontinuity (also called a discontinuity of...

with least element 0 and greatest element 1. The meet a ∧ b and the join a ∨ b of two elements coincide with their infimum and supremum, respectively, with...

closure under countable unions and intersections is paramount. Set limits are defined as follows on σ-algebras. The limit supremum or outer limit of a sequence...

the disk of radius r {\displaystyle r} and ‖ f ‖ ∞ , B r {\displaystyle \|f\|_{\infty ,B_{r}}} denotes the supremum norm of f ( z ) {\displaystyle f(z)}...

values between x and c. (The symbols inf and sup denote the infimum and supremum.) From the differentiability of f and g on I {\displaystyle {\mathcal {I}}}...

admissible decision rules. A minimax Bayes rule is one that minimises the supremum risk sup θ ∈ Θ R ( θ , d ∗ ) {\displaystyle \sup _{\theta \in \Theta }R(\theta...

A similar Galois connection whose lower adjoint is given by the meet (infimum) operation can be found in any Heyting algebra. Especially, it is present...

elements have a unique supremum (least upper bound) and an infimum (greatest lower bound), used in various areas of mathematics and logic. Laver 1.  Richard...

systems by universal and existential quantifiers. The semantics of the universal quantifier in t-norm fuzzy logics is the infimum of the truth degrees...