id: 05538827
dt: b
an: 2010b.00587
au: Athreya, S. R.; Sunder, V. S.
ti: Measure \& probability.
so: Hyderabad: Universities Press; Boca Raton, FL: CRC Press; (ISBN
978-1-4398-0126-0/hbk). x, 221~p. (2008).
py: 2008
pu: Hyderabad: Universities Press; Boca Raton, FL: CRC Press;
la: EN
cc: K55 K65 I55
ut: Bernoulli trials; Lebesgue measure; law of large numbers; central limit
theorem; Kolmogorov’s consistency theorem; Markov chain; Riesz’s
representation; Radon-Nikodym’s theorem; Stone-Weierstrass theorem
ci:
li:
ab: This is textbook on the measure, integration and probability basics written
for bachelor and master students in mathematics. The first chapter
deals with the fundamentals of measure theory, the Caratheodory
extension, the Lebesgue measure and its connection to Bernoulli trials.
In Chapter 2, the theory of Lebesgue integration is developed. Chapter
3 introduces probability laws, independence, and conditional
expectations. Probability measures on (infinite) product spaces and the
Kolmogorov consistency theorem are considered in Chapter 4. In Chapter
5 the main limit theorems of the probability theory are proven: the law
of large numbers and the central limit theorem. Chapter 6 is devoted to
the theory of discrete time Markov chains with a countable state space.
In the final Chapter 7 the authors study complex measures, $L^p$
spaces, Radon-Nikodym’s theorem, and the Riesz’s representation.
Each chapter is provided with numerous exercises. The attention of an
interested reader should be attracted to not too standard proofs of the
strong law of large numbers and the Riesz representation. Finally, the
Appendix contains basic information on metric and topological spaces,
compactness and the Stone-Weierstrass theorem.
rv: Ilya Pavlyukevich (Berlin)