A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence struct...

Using only elementary properties of the McShane integral for vector-valued functions, we establish a convergence theorem which for the scalar case of the integral yields the classical Beppo Levi (monotone) convergence theorem as an immediate corollary. As an application, the convergence theorem is used to prove that the space of McShane integrable functions, although not usually complete, is ul...

In this largely expository article, we highlight the significance of various types of ‘dimension’ for obtaining uniform convergence results in probability theory and we demonstrate how these results lead to certain notions of generalization for classes of binary-valued and realvalued functions. We also present new results on the generalization ability of certain types of artificial neural netwo...

The main purpose of this paper is to establish different types of convergence theorems for fuzzy Henstock integrable functions, introduced by  Wu and Gong cite{wu:hiff}. In fact, we have proved fuzzy uniform convergence theorem, convergence theorem for fuzzy uniform Henstock integrable functions and fuzzy monotone convergence theorem. Finally, a necessary and sufficient condition under which th...

We develop a general framework for the probabilistic analysis of random finite point clouds in context topological data analysis. extend notion barcode cloud to compact metric spaces. Such lives completion space barcodes with respect bottleneck distance, which is quite natural from an analytic view. As application we prove that i.i.d. variables sampled converge support their distribution when n...

We show there are no non-trivial finite Abelian group-valued measures on the lattice of closed subspaces of an infinite-dimensional Hilbert space, and we use this to establish that the unigroup of the lattice of closed subspaces of an infinite-dimensional Hilbert space is divisible. The main technique is a combinatorial construction of a set of vectors in R2n generalizing properties of those us...

Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in the univariate case and for the uniform distribution, the least-squares method is optimal in expectation in [1] and in probability in [7]...

The bounded convergence theorem on the Riesz space-valued Choquet integral is formalized for a sequence of measurable functions converging in measure and in distribution. 2010 Mathematics Subject Classification: Primary 28B15; Secondary 28A12, 28E10

Given a Hausdorff uniform space $$X$$ with the countable gage of pseudometrics uniformity , we introduce concept approximate variation function $$f$$ mapping subset $$T$$ reals into : this is infimum family Jordan-type variations all functions $$g:T\to X$$ which differ from in each pseudometric, generated by pseudometric gage, not greater than $$\varepsilon>0$$ . We prove following compactness ...