In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces . The concept captures the limiting behavior of finite configurations in the spaces employing an ultrafilter to bypass the need for repeatedly consideration of subsequences to ensure convergence. Ultralimits generalize Gromov–Hausdorff convergence in metric spaces.
An ultrafilter, denoted as ω, on the set of natural numbers is a set of nonempty subsets of (whose inclusion function can be thought of as a measure) which is closed under finite intersection, upwards-closed, and also which, given any subset X of , contains either X or \ X. An ultrafilter on is non-principal if it contains no finite set.
In the following, ω is a non-principal ultrafilter on .
If is a sequence of points in a metric space (X,d) and x∈ X, then the point x is called ω-limit of xn, denoted as , if for every it holds that
It is observed that,
A fundamental fact [1] states that, if (X,d) is compact and ω is a non-principal Ultrafilter on , the ω-limit of any sequence of points in X exists (and is necessarily unique).
In particular, any bounded sequence of real numbers has a well-defined ω-limit in , as closed intervals are compact.
Let ω be a non-principal ultrafilter on . Let (Xn ,dn) be a sequence of metric spaces with specified base-points pn ∈ Xn.
Suppose that a sequence , where xn ∈ Xn, is admissible. If the sequence of real numbers (dn(xn ,pn))n is bounded, that is, if there exists a positive real number C such that , then denote the set of all admissible sequences by .
It follows from the triangle inequality that for any two admissible sequences and the sequence (dn(xn,yn))n is bounded and hence there exists an ω-limit . One can define a relation on the set of all admissible sequences as follows. For , there is whenever This helps to show that is an equivalence relation on
The ultralimit with respect to ω of the sequence (Xn,dn, pn) is a metric space defined as follows. [2]
Written as a set, .
For two -equivalence classes of admissible sequences and , there is
This shows that is well-defined and that it is a metric on the set .
Denote .
Suppose that (Xn ,dn) is a sequence of metric spaces of uniformly bounded diameter, that is, there exists a real number C > 0 such that diam(Xn) ≤ C for every . Then for any choice pn of base-points in Xnevery sequence is admissible. Therefore, in this situation the choice of base-points does not have to be specified when defining an ultralimit, and the ultralimit depends only on (Xn,dn) and on ω but does not depend on the choice of a base-point sequence . In this case one writes .
Actually, by construction, the limit space is always complete, even when (Xn,dn) is a repeating sequence of a space (X,d) which is not complete. [5]
An important class of ultralimits are the so-called asymptotic cones of metric spaces. Let (X,d) be a metric space, let ω be a non-principal ultrafilter on and let pn ∈ X be a sequence of base-points. Then the ω–ultralimit of the sequence is called the asymptotic cone of X with respect to ω and and is denoted . One often takes the base-point sequence to be constant, pn = p for some p ∈ X; in this case the asymptotic cone does not depend on the choice of p ∈ X and is denoted by or just .
The notion of an asymptotic cone plays an important role in geometric group theory since asymptotic cones (or, more precisely, their topological types and bi-Lipschitz types) provide quasi-isometry invariants of metric spaces in general and of finitely generated groups in particular. [6] Asymptotic cones also turn out to be a useful tool in the study of relatively hyperbolic groups and their generalizations. [7]
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In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space. There is no general definition of an operator, but the term is often used in place of function when the domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly, and may be extended so as to act on related objects. See Operator (physics) for other examples.
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This article is supplemental for “Convergence of random variables” and provides proofs for selected results.
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