The Existence of Measurable Approximating Maximums

Certain statistical models are described by a parametrized family of sequences of random functions adapted to the fixed sequence of -algebras. The main problem is to estimate maximum points of the associated information function which is the limit of the sequence of random functions. For this various sequences of maximum functions are used. The main theorem of this paper establishes the existence of measurable approximating maximums associated to such a family which is indexed by the second countable Hausdorff space satisfying the Blackwell property. These spaces might be characterized as the second countable analytic ones. It is moreover shown that for separable families of sequences of adapted random functions the same theorem remains valid without Blackwell property as well.