Virtual continuity

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. We discuss virtually continuous functions and their applications to Kantorovich optimal transportation problem, Sobolev embedding theorems and operator theory.