Probabilistic convergence spaces.

*(English)*Zbl 0943.54002Introduction: Probabilistic metric and topological spaces have been studied extensively during the past thirty years; background information can be found in [M. J. Frank, J. Math. Anal. Appl. 34, 67-81 (1971; Zbl 0221.60001); B. Schweizer and A. Sklar, Probabilistic metric spaces (1983; Zbl 0546.60010); H. Sherwood, J. Lond. Math. Soc. 44, 441-448 (1969; Zbl 0167.46202)]. However, it is well known that many types of convergence which arise naturally in probability, statistics, and analysis are non-topological. In [Aequationes Math. 38, No. 2/3, 123-145 (1989; Zbl 0714.54002)] L. C. Florescu initiated a study of ‘probabilistic convergence structures’ based on net convergence. We believe that a more satisfactory theory can be developed in terms of filter convergence, and this paper gives an introduction to such a theory.

We define a probabilistic convergence structure on a set \(X\) to be a function \({\mathbf q}\) mapping \({\mathbf F}(X)\times I\) into \(2^X\), where \({\mathbf F}(X)\) is the set of all filters on \(X\), \(I\) is the closed interval \([0,1]\) in \(\mathbb{R}\), and \(2^X\) is the power set of \(X\); certain additional conditions are also imposed. Essentially, it is convenient to think of \({\mathbf q}\) as a family of convergence structures \(\{q_\lambda: \lambda\in I\}\); if a filter \({\mathcal F}\) \(q_\lambda\)-converges to \(x\), we say that ‘the probability that \({\mathcal F}\) \({\mathbf q}\)-converges to \(x\) is at least \(\lambda\)’. Thus \({\mathbf q}\) gives us a rule for determining the probability that any given filter on \(X\) converges to any given point in \(X\). These notions are made more precise in Section 2.

Section 1 begins with a review of some basic definitions, notations, and terminology pertaining to convergence spaces. We then investigate conditions \(K\) and \(F\) in the setting of convergence spaces. The former was introduced by H.-J. Kowalsky in [Math. Nachr. 12, 301-340 (1954; Zbl 0056.41403)], the latter by H. R. Fischer in some unpublished notes written around 1974. We answer a question raised by Fischer concerning the possible equivalence of these conditions by showing that a convergence space satisfying \(F\) is necessarily topological, whereas one satisfying \(K\) is generally not. We also show that \(K\), unlike \(F\), is not preserved under formation of initial structures (except in the special case where the generating functions are all injective).

In Section 2 we define probabilistic convergence spaces and extend such familiar convergence properties as regularity, first countability, and local compactness to such spaces. We also give three examples, two of which arise naturally from real analysis, which illustrate these properties. By considering ‘constant’ probabilistic convergence spaces, we observe that all of convergence space theory is included within the realm of probabilistic convergence space theory.

In Section 3, we extend the diagonal conditions \(K\) and \(F\) to a probabilistic convergence space equipped with a ‘\(t\)-norm’ \(T\) [see B. Schweizer and A. Sklar, loc. cit.]. If \((X, {\mathbf q}, T)\) satisfies the Kowalsky (respectively, Fischer) axiom, it is called a Kowalsky (respectively, Fischer) probabilistic convergence space. We show that every Fischer probabilistic convergence space is pretopological, and indeed for pretopological probabilistic convergence spaces the two axioms are equivalent (relative to any fixed \(t\)-norm). We show that Fischer probabilistic convergence spaces are not always topological except in the case when a certain restriction is placed on the \(t\)-norm. The same three examples studied in Section 2 are used in Section 3 to shed further light on the significance and applicability of the diagonal conditions.

We define a probabilistic convergence structure on a set \(X\) to be a function \({\mathbf q}\) mapping \({\mathbf F}(X)\times I\) into \(2^X\), where \({\mathbf F}(X)\) is the set of all filters on \(X\), \(I\) is the closed interval \([0,1]\) in \(\mathbb{R}\), and \(2^X\) is the power set of \(X\); certain additional conditions are also imposed. Essentially, it is convenient to think of \({\mathbf q}\) as a family of convergence structures \(\{q_\lambda: \lambda\in I\}\); if a filter \({\mathcal F}\) \(q_\lambda\)-converges to \(x\), we say that ‘the probability that \({\mathcal F}\) \({\mathbf q}\)-converges to \(x\) is at least \(\lambda\)’. Thus \({\mathbf q}\) gives us a rule for determining the probability that any given filter on \(X\) converges to any given point in \(X\). These notions are made more precise in Section 2.

Section 1 begins with a review of some basic definitions, notations, and terminology pertaining to convergence spaces. We then investigate conditions \(K\) and \(F\) in the setting of convergence spaces. The former was introduced by H.-J. Kowalsky in [Math. Nachr. 12, 301-340 (1954; Zbl 0056.41403)], the latter by H. R. Fischer in some unpublished notes written around 1974. We answer a question raised by Fischer concerning the possible equivalence of these conditions by showing that a convergence space satisfying \(F\) is necessarily topological, whereas one satisfying \(K\) is generally not. We also show that \(K\), unlike \(F\), is not preserved under formation of initial structures (except in the special case where the generating functions are all injective).

In Section 2 we define probabilistic convergence spaces and extend such familiar convergence properties as regularity, first countability, and local compactness to such spaces. We also give three examples, two of which arise naturally from real analysis, which illustrate these properties. By considering ‘constant’ probabilistic convergence spaces, we observe that all of convergence space theory is included within the realm of probabilistic convergence space theory.

In Section 3, we extend the diagonal conditions \(K\) and \(F\) to a probabilistic convergence space equipped with a ‘\(t\)-norm’ \(T\) [see B. Schweizer and A. Sklar, loc. cit.]. If \((X, {\mathbf q}, T)\) satisfies the Kowalsky (respectively, Fischer) axiom, it is called a Kowalsky (respectively, Fischer) probabilistic convergence space. We show that every Fischer probabilistic convergence space is pretopological, and indeed for pretopological probabilistic convergence spaces the two axioms are equivalent (relative to any fixed \(t\)-norm). We show that Fischer probabilistic convergence spaces are not always topological except in the case when a certain restriction is placed on the \(t\)-norm. The same three examples studied in Section 2 are used in Section 3 to shed further light on the significance and applicability of the diagonal conditions.

##### MSC:

54A20 | Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) |

54E70 | Probabilistic metric spaces |

60A05 | Axioms; other general questions in probability |