Unbounded convergence in the convergence vector lattices: a survey

Various convergences in vector lattices were historically a subject of deep investigation which stems from the begining of the 20th century in works of Riesz, Kantorovich, Nakano, Vulikh, Zanen, and many other mathematicians. The study of the unbounded order convergence had been initiated by Nakano in late 40th in connection with Birkhoff's ergodic theorem. The idea of Nakano was to define the almost everywhere convergence in terms of lattice operations without the direct use of measure theory. Many years later it was recognised that the unbounded order convergence is also rathe useful in probability theory. Since then, the idea of investigating of convergences by using their unbounded versions, have been exploited in several papers. For instance, unbounded convergences in vector lattices have attracted attention of many researchers in order to find new approaches to various problems of functional analysis, operator theory, variational calculus, theory of risk measures in mathematical finance, stochastic processes, etc. Some of those unbounded convergences, like unbounded norm convergence, unbounded multi-norm convergence, unbounded $\tau$-convergence are topological. Others are not topological in general, for example: the unbounded order convergence, the unbounded relative uniform convergence, various unbounded convergences in lattice-normed lattices, etc. Topological convergences are, as usual, more flexible for an investigation due to the compactness arguments, etc. The non-topological convergences are more complicated in genelal, as it can be seen on an example of the a.e-convergence. In the present paper we present recent developments in convergence vector lattices with emphasis on related unbounded convergences. Special attention is paid to the case of convergence in lattice multi pseudo normed vector lattices that generalizes most of cases which were discussed in the literature in the last 5 years.