Integrals and indicators of subharmonic functions. I

In the first part of our study, we consider general problems of the theory of density functions and $\rho$-semi-additive functions that are often used in the theory of growth of entire and subharmonic functions and in other branches of mathematics. In the theory of density functions, an important and often quoted theorem is the Polya theorem on the existence of a maximal and minimal density. The assertion 3 of the theorem 6 or the theorem 7 of the paper can be considered as the extension of the Polya theorem to a more general class of functions. The density functions have certain semi-additivity properties. Some problems of the theory of density functions and $\rho$-semi-additive functions are presented in the first part of our study. The central one here is the theorem 23, concerning the conditions for the existence at the zero of the derivative of $\rho$-semi-additivity function and estimation of integrals $ \int\limits_a^bf(t)\,d\nu(t) $ through the density functions of the function $\nu$. We note that the function $\nu$, in general, is not a distribution function of some countably-additive measure, and the integral must be understood as the Riemann-Stieltjes integral, and not as a Lebesgue integral in measure $\nu$.