Essential Range - Terminology and Useful Facts

Terminology and Useful Facts

• Throughout this article, the ordered pair (X, μ) will denote a measure space with non-negative additive measure μ.

• One property of non-negative additive measures is that they are monotone; that is if A is a subset of B, then μ(A) ≤ μ(B) if μ is additive.

• Let f be a function from a measure space (X, μ) to [0, ∞) and let S = { x | μ(ƒ−1((x, ∞))) = 0 }. The essential supremum of f, is defined to be the infimum of S. If S is empty, the essential supremum of f is defined to be infinity.

• If f is a function such that the essential supremum of g;; = |f| less than infinity, f is said to be essentially bounded.

• The vector space of all essentially bounded functions with the norm of a function defined to be its essential supremum, forms a complete metric space with the metric induced by its norm. Mathematically, this means that the collection of all essentially bounded functions form a Banach space. This Banach space is often referred to as L∞(μ) and is an Lp space.