Formal Definition
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 = { complex numbers z | μ({ x : abs(f(x) − z) < ε }) > 0 for all ε > 0}
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.
Read more about this topic: Essential Range
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