Let [Math Processing Error] be a set and let [Math Processing Error] be a ring of sets defined on [Math Processing Error]. A function mapping [Math Processing Error] to the set of positive real numbers [Math Processing Error] is called a pre-measure on [Math Processing Error], if [Math Processing Error] and if for every countable family of mutually disjoint subsets [Math Processing Error], [Math Processing Error], the measure of the union of these subsets equals the sum of the measures of each subset, formally: [Math Processing Error] The second property is called [Math Processing Error]-additivity.
Please note that the only difference between a pre-measure and a measure is that a pre-measure is defined on a ring of sets, while a measure is defined on a [Math Processing Error]-algebra.