On a Suffi cient Condition for Exhaustive Topological Measures
The focus of the work is centered on the question of relation between exhaustivity and countable additivity for functions of sets, given on the algebra of sets <i>B. </i>This algebra contains the system <i>F</i>, which is closed with respect to the fi nite sums and countable intersections of sets and which contains both empty set and the whole space <i>X</i>. A function of sets is called exhaustive whenever its values on a disjunctive sequence tend to zero. The following is the main result of the work. Theorem. Let function of sets possess the following density property: for any  > 0 and set <i>A </i> <i>B </i>there exists <i>K </i> <i>F </i>such that  (<i>AK </i>)< , where <i>AK </i>is the symmetric difference of sets. Let  be a dense measure on <i>B</i>. Then, if  is a countable additive measure on <i>B </i>then  is an exhaustive measure on <i>B.</i>