Rudin's Real and Complex Analysis - upto Section 1.4 review:
The chapter starts off fine with some motivation on why we need better integral than Riemman integral. Very nice observation in this section is - "The passage from Riemann's theory of integration to that Lesbegue is a process of completion...It is of same fundamental importance in analysis as is the construction of real number system from rationals".
Next section is a simple review of set theory. Then the concept of measurability is introduced while pointing out tconcept of measurability and continuity have some important properties in common.
Definition of measurability:
(i) \[X \in R\] So entire space is in collection.
(ii) if \[A \in R\] then \[A^c \in R\].
(iii) If \[A=\cup_{i=1}^{\infty} A_n\] and if \[A_n \in R\] for \[n=1,2,\cdots\] then \[A \in R\].
Just to show parallels, author also throws in definition of Topological space. But, then he never shows examples of measurable spaces. Perhaps he expects the readers to be familiar with these spaces. Perhaps...
Next section deals with review of metric spaces and topological definition of continuity. Plus he adds definition of continuity as a local property. Terseness of language in this book is fantastic! Definition of local continuity goes as follows:" A mapping $f$ of $X$ into $Y$ is said to be continuous at a point $x_0 \in X$ if for every neighborhood $V$ of $f(x_0)$, there corresponds a neighborhood $W$ of $x_0$, such that $f(W) \subset V$".
This proposition is clear enough (watch the beautiful, terse language!). Next section deals with comments on measurability which is totally out of place. This should have been presented after the definition of measurability as an exercise.
The chapter starts off fine with some motivation on why we need better integral than Riemman integral. Very nice observation in this section is - "The passage from Riemann's theory of integration to that Lesbegue is a process of completion...It is of same fundamental importance in analysis as is the construction of real number system from rationals".
Next section is a simple review of set theory. Then the concept of measurability is introduced while pointing out tconcept of measurability and continuity have some important properties in common.
Definition of measurability:
(i) \[X \in R\] So entire space is in collection.
(ii) if \[A \in R\] then \[A^c \in R\].
(iii) If \[A=\cup_{i=1}^{\infty} A_n\] and if \[A_n \in R\] for \[n=1,2,\cdots\] then \[A \in R\].
Just to show parallels, author also throws in definition of Topological space. But, then he never shows examples of measurable spaces. Perhaps he expects the readers to be familiar with these spaces. Perhaps...
Next section deals with review of metric spaces and topological definition of continuity. Plus he adds definition of continuity as a local property. Terseness of language in this book is fantastic! Definition of local continuity goes as follows:" A mapping $f$ of $X$ into $Y$ is said to be continuous at a point $x_0 \in X$ if for every neighborhood $V$ of $f(x_0)$, there corresponds a neighborhood $W$ of $x_0$, such that $f(W) \subset V$".
This proposition is clear enough (watch the beautiful, terse language!). Next section deals with comments on measurability which is totally out of place. This should have been presented after the definition of measurability as an exercise.
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