On generalization of mean value

Abstract

In paper we discuss the solution of mean value general form problem in case of all variables
symmetry absence. In 1930 A. N. Kolmogorov proved the formula for general form of mean
value. He formulated four axioms: continuity and monotony on each variable, symmetry on
each variable, mean value of equal variables is equal to these variables, any substitution of any
group of variables with their mean value does not change the mean value. In Kolmogorov’s
theorem all arguments are equitable, this means that the mean value is symmetric on each
variable. V. N. Chubarikov set the task of generalization to this result in case of all variables
symmetry absence. We divide all the variables on groups and the mean value is a symmetric
function for variables in each group separately. For example, if we have only one group the mean
value will be Kolmogorov’s mean value, so we have a generalization of Kolmogorov’s theorem.
In paper we show the general form of mean value in our case and we note the connection with
uniform distribution modulo 1.