Maps of several variables of finite total variation. I. Mixed differences and the total variation

Given two points a = ( a 1 , … , a n ) and b = ( b 1 , … , b n ) from R n with a < b componentwise and a map f from the rectangle I a b = [ a 1 , b 1 ] × ⋯ × [ a n , b n ] into a metric semigroup M = ( M , d , + ) , we study properties of the total variation TV ( f , I a b ) of f on I a b introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of bounded variation of several variables, in: Real Analysis Exchange 27th Summer Symposium, Opava, Czech Republic, 2003, pp. 217–222] such as the additivity, generalized triangle inequality and sequential lower semicontinuity. This extends the classical properties of C. Jordanʹs total variation ( n = 1 ) and the corresponding properties of the total variation in the sense of Hildebrandt [T.H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, 1963] ( n = 2 ) and Leonov [A.S. Leonov, On the total variation for functions of several variables and a multidimensional analog of Hellyʹs selection principle, Math. Notes 63 (1998) 61–71] ( n ∈ N ) for real-valued functions of n variables.