Main research fields
Maximum (minimum) / supremum (infimum) operators
·
the theory of integration in classical analysis,
·
the Lebesgue integral in
measure theory,
·
modern probability theory (cf. the laws large
numbers, say),
·
mathematical statistics (the
fundamental theorem due to Glivenko).
Measurable
functions
·
It is well known that measurable functions admit a
very simple structure. In the early 90’s my attention have been focused on the
question to know which are those properties of measurable
functions that cannot be treated by means of measure theory.
·
For example the boundedness
of measurable functions, the boundedness as well as
the asymptotical behaviours of sequences of
measurable functions (uniform, pointwise, and the Császár-Laczkovich (discrete, equally) types of convergence)
cannot.
·
However, these questions can be
answered by the means of optimal measure theory. The pioneer works can be
found in Acta Math. Hung. 63
(1-2) (1994), 1-15 and Publ.
Math.
What
is the totality of all the σ-algebras which are
equivalent to power sets?
Relations between Lebesgue
integral and optimal average.
Relations between probability
measure and optimal measure.
·
We should like to
point out that similar relations between probability measure and optimal
measure are available. Furthermore, the entropy and
alike also can be similarly bounded below and
above in terms of optimal
measure.
Applications