Convergence Theorems in the Theory of Integration

Abstract

This section gives a summary of some elementary facts used frequently throughout this book, and can be regarded as an appendix. In particular, we consider sufficient conditions for the interchange of integration and limit operations. In detail, we discuss a result on uniform convergence, the dominated convergence theorem, the bounded convergence theorem, Fatou’s lemma, and the monotone convergence theorem from the points of view of both Lebesgue integration theory and Riemann integration theory. Note that these are well-known results; hence we will be brief in details. For the proof of the monotone convergence theorem and Fubini’s theorem we merely refer to the appropriate literature.