ABSTRACT
This chapter develops the Daniell’s extension process. The idea behind the procedure is illustrated by the customary introduction of the Riemann integral in terms of the upper and lower Darboux sums. Riemann integrability can be characterized in terms of functionals on upper and lower functions. The chapter describes the upper and lower functions in Daniell’s sense and defines integrable functions. The chapter formulates convergence theorems to provide the foundations for applications of the theory of integration. It discusses the monotone convergence theorem, Fatou’s lemma, and Lebesgue convergence theorem. The chapter also proves two theorems related to the investigation of properties of integrable functions by using the induction principle.
