Main research fields

 

 

Maximum (minimum) / supremum (infimum) operators

 

  • The maximum (minimum) / supremum (infimum) operators rather constitute the core of analysis, measure theory, say.
  • They proved to be very much indispensable in many areas of mathematics. For example it is worth to mention that without maximum (minimum) / supremum (infimum) operators the following areas would fail to exist:

 

·         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. Debrecen 46 / 1-2 (1995), 79-87.  I. Fazekas – renaming a good deal of the results in the above works – showed (see, Publ. Math. Debrecen 51 / 3-4 (1997), 273-277) how to prove the structure theorem without the Zorn’s lemma, despite that he had written to a prestigious forum qualifying the theory of optimal measure as a balloon.

 

What is the totality of all the σ-algebras which are equivalent to power sets?

 

  • It is obvious that on a given non-empty set Ω every finite σ-algebra is equivalent to a power set. Thus it was natural to ask the question whether or not on a given infinite set Ω every infinite σ-algebra is equivalent to a power set. The answer to this question is deeply rooted in the theory of optimal measure (see Mathematical Notes, Miskolc, 2 / 2 (2001), 85 – 92).

 

 

Relations between Lebesgue integral and optimal average

 

  • Despite that the mathematical expectation is a linear operator and the optimal average a non-linear operator it is possible to find lower and upper bounds for the mathematical expectation using some groups of optimal averages.

 

 

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

 

  • Many areas in mathematics where additive or non-additive operators are considered can become application grounds for optimal average as well. In particular as Lebesgue integral can be ”collectivelyapproximated with optimal average, we convey that the electronic computation time of Lebesgue integral will considerably reduced.

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