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**Basic Analysis I**is a textbook by Jiří Lebl that covers the fundamentals of**real****analysis**, such as limits, continuity, differentiation, integration, and sequences and series. The book is suitable for undergraduate students and includes exercises and solutions. It is also available as a free**PDF**or a paperback on Amazon.1 Introduction. We begin by discussing the motivation for

**real analysis**, and especially for the reconsideration of the notion of integral and the invention of Lebesgue integration, which goes beyond the Riemannian integral familiar from clas- sical calculus. 1. Usefulness of**analysis**.honours undergraduate-level

**real****analysis**sequence at the Univer-sity of California, Los Angeles, in 2003. Among the undergradu-ates here,**real****analysis**was viewed as being one of the most dif-ﬂcult courses to learn, not only because of the abstract concepts being introduced for the ﬂrst time (e.g., topology, limits, mea-From here, there are some very important deﬁnitions in

**real****analysis**. We say that b 0 is the least upper bound,orthesupremumofEif A) b 0 isanupperboundforEand B) ifbisanupperboundforEthenb 0 b: Wedenotethisasb 0 = supE. Similarly,wesaythatc 0 isthegreatestlowerbound,ortheinﬁnimumofEif A) c 0 isalowerboundforEand B) ifcisalowerboundforEthenc ...found in a ﬁrst-year graduate course in

**real****analysis**. Although the presentation is based on a modern treatment of measure and integration, it has not lost sight of the fact that the theory of functions of one**real**variable is the core of the subject. It is assumed that the student has had a solid course in Advanced Calculus. Although0.2. ABOUT

**ANALYSIS**7 0.2 About**analysis****Analysis**is the branch of mathematics that deals with inequalities and limits. The present course deals with the most basic concepts in**analysis**. The goal of the course is to acquaint the reader with rigorous proofs in**analysis**and also to set a ﬁrm foundation for calculus of one variable (and several**Real****Analysis**Part I: MEASURE THEORY 1. Algebras of sets and σ-algebras For a subset A ⊂ X, the complement of A in X is written X −A. If the ambient space X is understood, in these notes we will sometimes write Ac for X −A. In the literature, the notation A′ is also used sometimes, and the textbook uses A˜ for the complement of A.