Essential range
In mathematics, particularly measure theory, the essential range of a function is intuitively the 'non-negligible' range of the function: It does not change between two functions that are equal almost everywhere. 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. Formal definition. Let "f" be a Borel-measurable, complex-valued function defined on a measure space formula_1. Then the essential range of "f" is defined  to be the set: In other words: 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. Extension. The notion of essential range can be extended to the case of formula_18, where formula_19 is a separable metric space. If formula_20 and formula_19 are differentiable manifolds of the same dimension, if formula_22 and if formula_23, then formula_24.