Negligible sets define several useful concepts that can be applied in various situations, such as truth almost everywhere.
The opposite of a negligible set is a generic property, which has various forms.
Let X be the set N of natural numbers, and let a subset of N be negligible if it is finite.
Let X be the set R of real numbers, and let a subset A of R be negligible if for each ε > 0,[1] there exists a finite or countable collection I1, I2, … of (possibly overlapping) intervals satisfying: and This is a special case of the preceding example, using Lebesgue measure, but described in elementary terms.
Let X be a directed set, and let a subset of X be negligible if it has an upper bound.
The first example is a special case of this using the usual ordering of N. In a coarse structure, the controlled sets are negligible.