A Nonstandard Characterization of Weak Convergence

2021

We establish a framework for the study of the effective theory of weak convergence of measures. We define two effective notions of weak convergence of measures on R: one uniform and one non-uniform. We show that these notions are equivalent. By means of this equivalence, we prove an effective version of the Portmanteau Theorem, which consists of multiple equivalent definitions of weak convergence of measures.

Topology and Its Applications, 2008

Pacific Journal of Mathematics, 1984

An integration theory for vector functions and operator-valued measures is outlined, and it is shown that in the setting of locally convex topological vector spaces, the dominated and bounded convergence theorems are almost equivalent to the countable additivity of the integrating measure. The measures studied are those representing the continuous linear operators on a space of continuous functions. When certain restrictions are imposed on the space involved, actual equivalence of countable additivity and the above theorems obtains, as well as equivalence of certain compactness properties of the operator being represented. An example is given which shows that, in general spaces, convergence in measure no longer implies the almost everywhere convergence of a subsequence.

2021

Abstract. A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the σ(L, L) topology. In this paper, we link such a result to weak convergence theory of bounded measures as exposed in Billingsley (1968) and in Lo(2021) to offer a detailed and new proof using the ideas beneath the proof of prohorov’s theorem where the continuous integrability replaces the uniform or asymptotic tightness.

Proceedings of the American Mathematical Society, 1995

This paper presents some properties of bounded linear functionals on a complete abstract M spaces, from which some criteria for weak convergence and weak compactness in such spaces are obtained.

Journal of Mathematical Analysis and Applications, 1986

We will use the concept of strong generating and a simple renorm-ing theorem to give new proofs to slight generalizations of some results of Argyros and Rosenthal on weakly compact sets in L 1 (µ) spaces for finite measures µ.

The notion of a strong measure zero space (but not this terminology) was introduced in [1] by E. Borel. A separable metric space X has strong measure zero if there is for each sequence ( n : n ∈ N) of positive real numbers a partition X = ∪ n X n such that for each n the diameter of X n is less than n . In we found characterizations of the notion of a strong measure zero set in terms of certain selection principles applied to open covers, and also in terms of Ramseyan partition relations for certain families of open covers. Also, the property of having strong measure zero in all finite powers was characterized like this. Due to successes in using the concept of a filter on the set of natural numbers to describe certain covering properties of sets of real numbers, as in , and of using the closure properties of function spaces, as in , the question arose whether these methods would also yield analogous descriptions of strong measure zero spaces. In the paper [9] we made a beginning in this direction by showing that for certain function spaces derived from a space X, the strong measure zero-ness of all finite powers of X is equivalent to certain closure properties of the associated function space. The function space associated with X was obtained by first associating with X a subspace T (X) of the Alexandroff double of the closed unit interval [0,1], and by then taking the function space C s (T (X)) -this space differs from the usual space topologized by the topology of pointwise convergence in that we consider a coarser topology described in terms of a dense discrete subspace consisting of the isolated points of T (X).

Involve, a Journal of Mathematics, 2017

For a topological space X, it is a natural undertaking to compare its topology with the weak topology generated by a family of real-valued continuous functions on X. We present a necessary and sufficient condition for the coincidence of these topologies for an arbitrary family A ⊂ C(X). As a corollary, we give a new proof of the fact that families of functions which separate points on a compact space induce topologies that coincide with the original topology.