THE STRUCTURE OF THE CLASS OF INFINITETY DIVISIBLE POSITIVE P-FUNCTIONS

The studying of the non-standard p-functions is so far still a remarkable topic. In this paper it is proved that the class X of measurable positive p-functions is a hereditary subsomigroup of the semigroup X of p-functions, and that the class X^I ofinfinitely divisible positive p-functions which is identical with the class of bounded increasing-ratio functions is a subsemigroup of X. And more, X^I=δy^I where δ is the class of constant functions with values in(0, 1, X^I is the class of infinitely divisible standard p-functions. Furthermore, in this paper, is slved completely the problem of the structure of the I_o-class y^I_o of positive p-functions, i. e., y^I_o=X, where \hat\delta is the class of bounded exponential functions. This result generalizes the exciting one about the I_o-class of standard p-functions, y^I_o=\hat\delta, obtained by Kendall, Davidson and ynahobcknn by long-term studying.