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Concept
Negligible set
Summary
In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose.
As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function.
Negligible sets define several useful concepts that can be applied in various situations, such as truth almost everywhere.
In order for these to work, it is generally only necessary that the negligible sets form an ideal; that is, that the empty set be negligible, the union of two negligible sets be negligible, and any subset of a negligible set be negligible.
For some purposes, we also need this ideal to be a sigma-ideal, so that countable unions of negligible sets are also negligible.
If I and J are both ideals of subsets of the same set X, then one may speak of I-negligible and J-negligible subsets.
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.
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.
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