Negligible set

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,[1] there exists a finite or countable collection I₁, I₂, … of (possibly overlapping) intervals satisfying:

${\displaystyle A\subset \bigcup _{k}I_{k}}$

and

${\displaystyle \sum _{k}|I_{k}|<\epsilon .}$

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.

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.

Let X be a set, and let I be an ideal of negligible subsets of X. If p is a proposition about the elements of X, then p is true almost everywhere if the set of points where p is true is the complement of a negligible set. That is, p may not always be true, but it's false so rarely that this can be ignored for the purposes at hand.

If f and g are functions from X to the same space Y, then f and g are equivalent if they are equal almost everywhere. To make the introductory paragraph precise, then, let X be N, and let the negligible sets be the finite sets. Then f and g are sequences. If Y is a topological space, then f and g have the same limit, or both have none. (When you generalise this to a directed sets, you get the same result, but for nets.) Or, let X be a measure space, and let negligible sets be the null sets. If Y is the real line R, then either f and g have the same integral, or neither integral is defined.

• Negligible function
• Generic property