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.
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