Statistical convergence in probabilistic generalized metric spaces w r.t. strong topology Full Text

Requiring that a sequence of distances tends to zero is a standard criterion for convergencein a metric space. In the following, according to the generalization of asymptotic density given in , statistically convergent and Cauchy sequences in a PGM-space are introduced. In this section, some basic definitions and results related to PM-space, PGM-space, and statistical convergence are presented and discussed. First, recall the definition of triangular norm (t-norm) as follows. Values coincide exactly with the statistical convergence in PM-space and PGM-space (related to G-metric), respectively.

Similarly the continuum of a plane is viewed as simply a collection of points. And the space within a sphere or other solid figure is also viewed as a collection of points. The validity of viewing a continuum as simply a collection of points is not at all obvious to me. I think the validity of doing it would be questioned by anyone first introduced to the idea. Provides a measure of how different those two realizations are. The following corollary is a direct consequence of the above theorem.

2 Cauchy Criterion for Convergence

We have illustrated with an example of points in 3-space but the sequence could also, for example, be a sequence of functions in a function space. We can also let M be the set of all points in the plane. 12 shows typical open, closed and general sets in the plane. A point P is called a limit point of a point set S if every ε-deleted neighborhood of P contains points of S. We see, from the definitions, that while an ε-neighborhood of a point is an open set a neighborhood of a point may be open, closed or neither open nor closed.. The following example illustrates the concept of mean-square convergence.

11 is depicted a typical open set, closed set and general set on the real line. The set π corresponds to all possible unions and intersections of general sets in M. The union or intersection of any two open sets in M is open. Thus the collection of all open sets in M form a closed system with respect to the operations of union and intersection. They constitute a subset τ of the collection of all possible sets π in M. The concept of distance is intricately tied to the concept of a continuum of points.

How to measure similarity

The precise definition depends on what sort of space XX is. The notion is of particular and historical importance in analysis, where it serves to define for instance the notion of derivative. We now prove a generalization of the Extreme value theorem 5.3.7.

The space of continuous functions \(C()\) on the interval \(\) with sup-norm is a Banach space. Below is a characterization of closed sets via sequences. If \(\) is a Cauchy sequence and if \(\) has a convergent subsequence then \(\) converges. As before, this implies convergence of integrals against bounded measurable functions, but this time convergence is uniform over all functions bounded by any fixed constant. To formalize this requires a careful specification of the set of functions under consideration and how uniform the convergence should be.

Theorem 3.13

A metric space \(M\) is called complete if every Cauchy sequence in \(M\) converges in \(M\). Note that a complete vector space with a norm is called a Banach space. A Hilbert space is, therefore, a Banach space with a norm defined by the inner product. In a measure theoretical or probabilistic context setwise convergence is often referred to as strong convergence . This can lead to some ambiguity because in functional analysis, strong convergence usually refers to convergence with respect to a norm.

Every statistically convergent sequence in a PGM-space is statistically Cauchy. Every convergent sequence in a PGM-space is statistically convergent. The theory of probabilistic metric space (PM-space) as a generalization of ordinary metric space was introduced https://globalcloudteam.com/ by Menger in . In this space, distribution functions are considered as the distance of a pair of points in statistics rather than deterministic. Here the supremum is taken over f ranging over the set of all measurable functions from X to [−1, 1].

The Definition of a Metric Space

A set is closed when it contains the limits of its convergent sequences. Let X be a metric space, let Y be a complete metric space, and let A be a dense subspace of X. If f is a uniformly continuous mapping of A into Y, then f can be extended uniquely to a uniformly continuous mapping g of X into Y. The collection τ of all open sets in a metric space M doesn’t represent all possible sets that can be formed on M. Let π be the set of all possible sets that can be formed on M. The union or intersection of any two sets in π is a set in π.

In 2008, Sencimen and Pehlivan introduced the concepts of statistically convergent sequence and statistically Cauchy sequence in the probabilistic metric space endowed with strong topology. More generally than sequences, and equivalently to nets, we may speak of limits of filters on XX. This concept is axiomatized directly in the concept of convergence space. In the what is convergence metric case of a topological space XX, a filter of subsets of XX converges to a point xx if every neighbourhood of xx is contained in the filter. This result shows that continuous mappings of one metric space into another are precisely those which send convergent sequences into convergent sequences. In other words, they are those mappings which preserve convergence.

ProofWiki.org

If a distance concept exists that is similar to the familiar distance from three dimensional space, then a continuum of distances will exist. If a distance concept doesn’t exist, a continuum concept can’t exist. In a metric space, a sequence $$ can only converge to one limit, denoted by $\lim_x_n$. A metric space \(M\) is compact if and only if every sequence in \(M\) has a convergent subsequence. There are several equivalent definitions of weak convergence of a sequence of measures, some of which are more general than others. The equivalence of these conditions is sometimes known as the Portmanteau theorem.

• A Hilbert space is, therefore, a Banach space with a norm defined by the inner product.
• Is a sequence of probability measures on a Polish space.
• 0 Difference in the definitions of cauchy sequence in Real Sequence and in Metric space.
• In the following, some basic concepts of statistical convergence are discussed.
• But what if the sequence does not have a limit, or the limit is not known?
• The cover \(\_\) is called an open cover if each set \(U_i\) is open.

Perform an internet search and find dense subsets of \((C(), \norm_\infty)\) . Show that \(C()\) is a closed subset of \(\mathcal()\). Prove that an open ball \(B_\eps\subset M\) is open.

Completeness

The above properties correspond to certain central properties of distances in three dimensional Euclidean space. The distance d that is defined between “points” x and y of a metric space is called a metric or distance function. Use Theorem 3.3 to prove that the composition of continuous functions between metric spaces is continuous.