On the necessary and sufficient conditions for the measurability of a positive sequence

The work is devoted to finding out the necessary and sufficient conditions for the measurability of a sequence of positive numbers. The concept of logarithmic measurability of a sequence is also introduced. It is assumed that the considered sequences form a sequence of zeros of some entire function of exponential type. Therefore, clarification of this question can be useful in solving the problem of completeness of the system of exponents or exponential monomials in some convex domain. Such characteristics of the sequence as lower and upper densities, minimum and maximum densities, lower and upper logarithmic block densities play an important role.