Measure, Integration and a Primer on Probability Theory: Volume 1

✍ Scribed by Stefano Gentili

• Publisher: Springer International Publishing;Springer
• Year: 2020
• Tongue: English
• Leaves: 458
• Series: UNITEXT 125
• Edition: 1st ed.
• Category: Library

✦ Synopsis

The text contains detailed and complete proofs and includes instructive historical introductions to key chapters. These serve to illustrate the hurdles faced by the scholars that developed the theory, and allow the novice to approach the subject from a wider angle, thus appreciating the human side of major figures in Mathematics. The style in which topics are addressed, albeit informal, always maintains a rigorous character. The attention placed in the careful layout of the logical steps of proofs, the abundant examples and the supplementary remarks disseminated throughout all contribute to render the reading pleasant and facilitate the learning process. The exposition is particularly suitable for students of Mathematics, Physics, Engineering and Statistics, besides providing the foundation essential for the study of Probability Theory and many branches of Applied Mathematics, including the Analysis of Financial Markets and other areas of Financial Engineering.

✦ Table of Contents

Front Matter ....Pages i-xi
Front Matter ....Pages 1-1
Round-Up of Topology (Stefano Gentili)....Pages 3-12
Types of Sets (Stefano Gentili)....Pages 13-51
Front Matter ....Pages 53-53
Borel sets in (\mathbb {R}) (Stefano Gentili)....Pages 55-83
Baire Functions on (\mathbb {R}) (Stefano Gentili)....Pages 85-103
Borel Functions and Baire Functions (Stefano Gentili)....Pages 105-117
Front Matter ....Pages 3-119
Semi-algebra and Algebra of Sets (Stefano Gentili)....Pages 121-129
Monotone Classes and (\sigma )-Algebras (Stefano Gentili)....Pages 131-145
Front Matter ....Pages 147-166
Set Functions and Measure (Stefano Gentili)....Pages 167-196
The Lebesgue Measure (Stefano Gentili)....Pages 197-240
Measurable Functions (Stefano Gentili)....Pages 241-280
Front Matter ....Pages 281-284
The Lebesgue Integral (Stefano Gentili)....Pages 285-321
Comparing Notions of Integral (Stefano Gentili)....Pages 323-346
Front Matter ....Pages 347-353
Functions with Bounded Variation and Absolutely Continuous Functions (Stefano Gentili)....Pages 355-381
Fundamental Theorems of Calculus for the Lebesgue Integral (Stefano Gentili)....Pages 383-427
Back Matter ....Pages 429-463

✦ Subjects

Mathematics; Measure and Integration; Fourier Analysis; Functional Analysis; Probability Theory and Stochastic Processes