associated a real number d(p;q) called the distance from pto qsuch that 1. d(p;q) >0 if p6= q, and d(p;q) = 0 if p= q; 2. d(p;q) = d(q;p); 3. d(p;q) d(p;r)+d(r;q) for any r2X Any function with these three properties is called a distance function or metric .
Real Analysis Course Notes C. McMullen Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Set Theory and the Real Numbers . . . . . . . . . . . . . . . 4 3 Lebesgue Measurable Sets . . . . . . . . . . . . . . . . . . . . 13 4 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . 26
The study of real analysis is indispensible for a prospective graduate student of pure or applied mathematics.
Theorem 1.7. Every nonempty set of real numbers that is bounded from above has a supremum. Since inf A= −sup(−A), it follows immediately that every nonempty set of real numbers that is bounded from below has an infimum. Example 1.8. The supremum of the set of real numbers A= {x∈ R : x< √ 2} is supA= √ 2. By contrast, since √
6.1 BasicDefinitions. . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.2 UnilateralLimits. . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.3 Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4 UnilateralContinuity. . . . . . . . . . . . . . . . . . . . . . . . 108 6.5 ContinuousFunctions. . . . . . . ....
1.2 The set of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3 Absolute value and bounded functions . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4 Intervals and the size of R
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