Measure

We recollect the essential concepts and results from Measure Theory, mostly without proofs. We state the Monotone class theorem, Carathéodory’s extension theorem, and the Borel–Cantelli lemma. Essential results on integration, such as Fatou’s lemma, Monotone convergence theorem, Dominated convergence theorem, Fubini’s theorem, and some standard inequalities are given without proof. Some of the more important results such as the construction of Lebesgue–Stieltjes measures on \(\mathbb {R}^n\) , Jordan-Hahn decomposition, Radon–Nikodym theorem and the Lebesgue decomposition theorem are developed in details. The construction of the Lebesgue–Stieltjes measures is given in some details. Readers who need more details, can refer to one of many well-known books available.