Construction of Measure from Semialgebra of Sets1

Summary In our previous article [22], we showed complete additivity as a condition for extension of a measure. However, this condition premised the existence of a σ -field and the measure on it. In general, the existence of the measure on σ -field is not obvious. On the other hand, the proof of existence of a measure on a semialgebra is easier than in the case of a σ -field. Therefore, in this article we define a measure ( pre-measure ) on a semialgebra and extend it to a measure on a σ -field. Furthermore, we give a σ -measure as an extension of the measure on a σ -field. We follow [24], [10], and [31].