Construction of measures
Shmuel Kantorovitz, Ami Viselter · 2022
Abstract In this chapter we introduce a powerful technique due to Caratheodory for constructing positive measures from primitive objects called semi-measures on semi-algebras. In contrast to σ-algebras, which are normally “big” and intangible, semi-algebras are often “small” and concrete. We use this method to construct the Lebesgue–Stieltjes measures, a special case of which is the Lebesgue measure, which is roughly the “length measure” on the real line, and it covers the structure of measurable sets. We apply Caratheodory’s extension theorem to construct the product of two positive measure spaces and prove the theorems of Fubini and Tonelli. Finally, the chapter offer exercises to challenge the reader.