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Virtual continuity of measurable functions and its applications. (English. Russian original) Zbl 1320.28003

Russ. Math. Surv. 69, No. 6, 1031-1063 (2014); translation from Usp. Mat. Nauk 69, No. 6, 81-114 (2014).
The authors consider (probability) measure spaces isomorphic to the unit interval endowed with Lebesgue measure. It is a classical result that a measurable function on the unit interval is nearly continuous: for every \(\varepsilon>0\) there is a compact set of measure larger than \(1-\varepsilon\) on which the function is continuous. The theorem, naturally, also hold for the unit square but not in the form one would maybe expect: the characteristic function of \(\{(x,y):x\leq y\}\) shows that the compact set can not always be taken to be a product set. The authors define a function of two variables to be properly virtually continuous if one can find product sets of measure arbitrarily close to \(1\) on which the function is continuous and virtually continuous if it equal to a properly virtually continuous function on a set of full measure. In the case of general measure spaces the continuity is with respect to suitable (semi)metrics defined on the factors of the product set, hence these vary with \(\varepsilon\) as well. The bulk of the paper consists of a thorough study of these notions. The final section establishes virtual continuity of functions in various Sobolev spaces and of kernels of certain nuclear operators.
Reviewer: K. P. Hart (Delft)

MSC:

28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
26B05 Continuity and differentiation questions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
54E35 Metric spaces, metrizability
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