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.
Read more about this topic: Essential Range
Famous quotes containing the word facts:
“The history of his present majesty, is a history of unremitting injuries and usurpations ... all of which have in direct object the establishment of an absolute tyranny over these states. To prove this, let facts be submitted to a candid world, for the truth of which we pledge a faith yet unsullied by falsehood.”
—Thomas Jefferson (1743–1826)