Integration of Integrable Measures

This chapter defines the integral over a measurable set ω of an integrable measure f with values in a Neumann space E. This integral is the continuous extension, with respect to f and ω, of the Cauchy integral of a uniformly continuous function over an open set. The chapter shows that the integral is linear and provides a bound for the semi-norms of the integral. Continuity and linearity together imply uniform continuity. Continuity and linearity together imply uniform continuity. The chapter expresses the measure of a bounded measurable set in terms of an integral. It also defines positive real measures in order to state the monotonicity properties of the real-valued integral. The chapter also shows that the integral of a real-valued positive integrable measure increases with the domain of integration.