Negligible set
Encyclopedia
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 set
s 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 subset
s 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.
s, 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 pathalogical.
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.
s 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 net
s.)
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
, and null set
Null set
In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
s can be ignored when studying the integral of a measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
.
Negligible sets define several useful concepts that can be applied in various situations, such as truth almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
.
In order for these to work, it is generally only necessary that the negligible sets form an ideal
Ideal (set theory)
In the mathematical field of set theory, an ideal is a collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal , and the union of any two elements of the ideal must also be in the ideal.More formally, given a set X, an...
; that is, that the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
be negligible, the union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of two negligible sets be negligible, and any subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of a negligible set be negligible.
For some purposes, we also need this ideal to be a sigma-ideal
Sigma-ideal
In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is perhaps in probability theory.Let be a measurable space...
, so that countable unions of negligible sets are also negligible.
If I and J are both ideals of subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s 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
Generic property
In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic...
, which has various forms.
Examples
Let X be the set N of natural numberNatural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, and let a subset of N be negligible if
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
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
Uncountable set
In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...
, and let a subset of X be negligible if it is countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
.
Then the negligible sets form a sigma-ideal.
Let X be a measurable space equipped with a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
m, and let a subset of X be negligible if it is m-null
Null set
In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
.
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 pathalogical.
Let X be a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, 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
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in any open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
).
Then the negligible sets form a sigma-ideal.
X is a Baire space
Baire space
In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
if the interior
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
of every such negligible set is empty.
Let X be a directed set
Directed set
In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
, and let a subset of X be negligible if it has an upper bound
Upper bound
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
.
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
Coarse structure
In the mathematical fields of geometry and topology, a coarse structure on a set X is a collection of subsets of the cartesian product X × X with certain properties which allow the large-scale structure of metric spaces and topological spaces to be defined.The concern of traditional geometry and...
, the controlled sets are negligible.
Derived concepts
Let X be a set, and let I be an ideal of negligible subsetSubset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s of X.
If p is a proposition about the elements of X, then p is true almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
if the set of points where p is true is the complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
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
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, 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 net
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
s.)
Or, let X be a measure space, and let negligible sets be the null sets.
If Y is the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R, then either f and g have the same integral, or neither integral is defined.