Negligible Set - Examples

Examples

Let X be the set N of natural numbers, and let a subset of N be negligible if it is finite. Then the negligible sets form an ideal. This idea can be applied to any infinite set; but if applied to a finite set, every subset will be negligible, which is not a very useful notion.

Or let X be an uncountable set, and let a subset of X be negligible if it is countable. Then the negligible sets form a sigma-ideal.

Let X be a measurable space equipped with a measure m, and let a subset of X be negligible if it is m-null. Then the negligible sets form a sigma-ideal. Every sigma-ideal on X can be recovered in this way by placing a suitable measure on X, although the measure may be rather pathological.

Let X be the set R of real numbers, and let a subset A of R be negligible if for each ε > 0, 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 topological space, and let a subset be negligible if it is of first category, that is, if it is a countable union of nowhere-dense sets (where a set is nowhere-dense if it is not dense in any open set). Then the negligible sets form a sigma-ideal. X is a Baire space if the interior of every such negligible set is empty.

Let X be a directed set, and let a subset of X be negligible if it has an upper bound. Then the negligible sets form an ideal. The first example is a special case of this using the usual ordering of N.

In a coarse structure, the controlled sets are negligible.