TY - JOUR

T1 - Virtual continuity of measurable functions and its applications

AU - Vershik, A.M.

AU - Zatitskiy, P.B.

AU - Petrov, F.V.

PY - 2014

Y1 - 2014

N2 - © 2014 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd. A classical theorem of Luzin states that a measurable function of one real variable is 'almost' continuous. For measurable functions of several variables the analogous statement (continuity on a product of sets having almost full measure) does not hold in general. The search for a correct analogue of Luzin's theorem leads to a notion of virtually continuous functions of several variables. This apparently new notion implicitly appears in the statements of embedding theorems and trace theorems for Sobolev spaces. In fact it reveals the nature of such theorems as statements about virtual continuity. The authors' results imply that under the conditions of Sobolev theorems there is a well-defined integration of a function with respect to a wide class of singular measures, including measures concentrated on submanifolds. The notion of virtual continuity is also used for the classification of measurable functions of several variables

AB - © 2014 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd. A classical theorem of Luzin states that a measurable function of one real variable is 'almost' continuous. For measurable functions of several variables the analogous statement (continuity on a product of sets having almost full measure) does not hold in general. The search for a correct analogue of Luzin's theorem leads to a notion of virtually continuous functions of several variables. This apparently new notion implicitly appears in the statements of embedding theorems and trace theorems for Sobolev spaces. In fact it reveals the nature of such theorems as statements about virtual continuity. The authors' results imply that under the conditions of Sobolev theorems there is a well-defined integration of a function with respect to a wide class of singular measures, including measures concentrated on submanifolds. The notion of virtual continuity is also used for the classification of measurable functions of several variables

U2 - 10.1070/RM2014v069n06ABEH004927

DO - 10.1070/RM2014v069n06ABEH004927

M3 - Article

VL - 69

SP - 1031

EP - 1063

JO - Russian Mathematical Surveys

JF - Russian Mathematical Surveys

SN - 0036-0279

IS - 6

ER -