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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.
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 setIn mathematics, an uncountable set is a set which is not countable. Here, "countable" means countably infinite or finite, so by definition, all uncountable sets are infinite. The best known example of an uncountable set is the set R of all real numbers; C, and let a subset of X be negligible if it is countableIn mathematics the term countable set is used to describe the size of a set, e. the number of elements it contains. Non-mathematicians can usually only measure the size of finite sets by counting and have an unclear concept of infinite sets and the differ. Then the negligible sets form a sigma-ideal.
Let X be a measurable space equipped with a measureMeasure theory In mathematics, a measure is a function that assigns a number, e. a "size", "volume", or "probability", to subsets of a given set. The concept is important in mathematical analysis and probability theory. Measure theory is that branch of re 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.
Let X be a topological spaceTopological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies, 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.