Properties and Examples
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.
Read more about this topic: Essential Range
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