L p (Ω; E ) Spaces

This chapter constructs the Lebesgue space L p (Ω;E), and shows that it is sequentially complete which corresponds, for real-valued measures, to the Riesz-Fischer completeness theorem. It defines p -integrable measure with values in a Neumann space. One may define integrable distributions by requiring that the p -integrabilizing sequence converges only in the sense of distributions, rather than in the sense of measures. Construction via completion avoids the study of measurability. Once the measures are known, and they are indispensable to use, they provide a framework that is very well-suited to studying integrability.