Elements of Measure Theory and \(L^{p}\) Spaces

In this chapter we set forth the basic concepts of measure theory and develop the theory of integration on abstract measure spaces, paying particular attention to the Lebesgue integral on the Euclidean space \(\textbf{R}^{n}\) . In particular, we prove Minkowski’s inequality for integrals (Theorem 3.29) and Hardy’s inequality (Theorem 3.31) in \(L^{p}\) spaces. In Sect. 3.9, we prove the Marcinkiewicz interpolation theorem (Theorem 3.47) which plays an important role in the proof of Theorem 9.3 in Sect. 9.2 . In Sect. 3.10, as an application of Marcinkiewicz’s interpolation theorem we study Riesz potentials in the classical potential theory (Theorem 3.48). This chapter is included for the sake of completeness, to minimize the necessity of consulting too many outside references. This makes the monograph fairly self-contained.