Essential range

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In mathematics, particularly measure theory, the essential range of a function is intuitively the 'non-negligible' range of the function. One way of thinking of the essential range of a function is the set on which the range of the function is most 'concentrated'. The essential range can be defined for measurable real or complex-valued functions on a measure space.

Terminology and useful facts[edit]

  • 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.

Formal definition[edit]

Let f be a complex-valued function defined on a measure space, (X, μ) that also belongs to L(μ). Then the essential range of f is defined to be the set:

S = \left\{z \in \mathbb{C} : \mu(\{x : |f(x) - z| < \varepsilon\}) > 0\ \text{for all}\ \varepsilon > 0\right\}

Note that: Another description of the essential range of a function is as follows:

The essential range of a complex-valued function is the set of all complex numbers z such that the inverse image of each ε-neighbourhood of z under f has positive measure.

The above description of the essential range is equivalent to the formal definition of the essential range and will therefore be used throughout this article.

Properties and examples[edit]

1. Every complex-valued function defined on the measure space (X, μ) whose absolute value is bounded, is essentially bounded. A proof is provided in the next section.

2. The essential range of an essentially bounded function f is always compact. The proof is given in the next section.

3. The essential range, S, of a function is always a subset of the closure of A where A is the range of the function. This follows from the fact that if w is not in the closure of A, there is a ε-neighbourhood, Vε, of w that doesn't intersect A; then f−1(Vε) has 0 measure which implies that w cannot be an element of S.

4. Note that the essential range of a function may be empty even if the range of the function is non-empty. If we let Q be the set of all rational numbers and let T be the power set of Q, then (Q, T, m) form a measurable space with T the sigma algebra on Q, and m a measure defined on Q that maps every member of T onto 0. If f is a function that maps Q onto the set of all points with rational co-ordinates that lie within the unit circle, then f has nonempty range (clearly). The essential range of f however is empty for if w is any complex number and V any ε-neighbourhood of w, then f−1(V) has 0 measure by construction.

5. Example 4 also illustrates that even though the essential range of a function is a subset of the closure of the range of that function, equality of the two sets need not hold.

Theorems[edit]

Theorem 1

Every bounded complex-valued function defined on (X, μ) is essentially bounded.

Proof:

If |f| is bounded, then |f| < a for some a > 0 so that if g = |f|, then g−1(a, ∞) is empty and therefore has measure 0. This implies that the set S = { x | μ(g−1((x, ∞))) = 0 } is nonempty so that the essential supremum of g is less than infinity. Therefore, f is essentially bounded.

Theorem 2

The essential range of a complex-valued function, f, defined on a measure space (X, μ) that belongs to L(μ) is compact if μ is a non-negative additive measure.

Proof

Let S denote the essential range of the function in question. By the Heine–Borel theorem, it suffices to show that S is closed and bounded. To show that S is closed, we will show that every convergent sequence in S converges to an element in S. Let (wn) be a convergent sequence of points in S and let w be its limit. Let V be an ε-neighbourhood of w; we will prove that the inverse image of Vε under f has positive measure. First of all, choose N such that n > Nwn belongs to Vε. Since Vε is open and since wN+1 belongs to Vε, we may choose a δ-neighbourhood, Vδ about wN+1 that is contained in Vε. Since wN+1 belongs to S, the inverse image of Vδ under f has positive measure. Since Vδ is a subset of Vε, f−1(Vδ) is a subset of f−1(Vε). Noting that f−1(Vδ) has positive measure, it follows that f−1(Vε) has positive measure. Since ε was arbitrary, it follows that w belongs to S and S is closed.

Note that since f is essentially bounded, there exists a such that g(a, +infinity) has 0 measure where g = |f|. Therefore, if w is a complex number such that |w| > a, and K = {complex numbers z | |z| > a}, then there is a p-neighbourhood, Vp, of w that is contained in K (since K is open). Note that g−1(a, +infinity) = f−1(K) so that f−1(K) has 0 measure. If f−1(Vp) had positive measure, so would f−1(K) since f−1(Vp) is a subset of f−1(K); a contradiction. Therefore, f−1(Vp) has 0 measure so that w cannot be an element of S. This shows that S is a subset of the complement of K so that S is bounded.

Applications of the theorems and additional notes

1. Note that the essential range of a function always lies within a closed ball in R2 of radius equal to the essential supremum of the function.

2. An essentially bounded function is intuitively a function that is unbounded on a set of measure 0, i.e. a negligible set in a measure-theoretic sense. A bounded function is basically a function that is unbounded on the empty set (which is not mathematically precise but gives the basic idea). Since the empty set has measure 0, one can believe that every bounded function is essentially bounded. This fact is proven in theorem 1.

3. Note that the proof of theorem 2 is largely dependent on the fact that non-negative additive measures are monotone.

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