Ultralimit

In mathematics, an ultra limit is a geometric construction that assigns a limit metric space to a sequence of metric spaces . The concept of such captures the limiting behavior of finite configurations in the spaces and employs an ultrafilter to bypass the need for repeatedly considering subsequences to ensure convergence. Ultra limits generalize the idea of 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 the ω-limit of xn, denoted as . If for every there are: It is observed that, If an ω-limit of a sequence of points exists, it is unique. If in the standard sense, . (For this property to hold, it is crucial that the ultrafilter shoul be non-principal.) A fundamental fact 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.