id: 06151688
dt: b
an: 2014a.00725
au: Ovchinnikov, Sergei
ti: Measure, integral, derivative. A course on Lebesgue’s theory.
so: Universitext. New York, NY: Springer (ISBN 978-1-4614-7195-0/pbk;
978-1-4614-7196-7/ebook). x, 146~p. (2013).
py: 2013
pu: New York, NY: Springer
la: EN
cc: I55
ut: Lebesgue measure; Lebesgue integral; differentiation
ci:
li: doi:10.1007/978-1-4614-7196-7
ab: The monograph is intended for a one semester course on Lebesgue’s theory
and deals with measure, integration and differentiation. It is
accessible to upper-undergraduate and lower graduate level students,
and the only prerequisite is a course in elementary real analysis. The
book contains four chapters and an appendix, the first chapter being
devoted to some preliminaries that intend to fill the gap between what
the student may have learned before and what is required to understand
this text; and the appendix being devoted to remedy a limitation of the
book that, in the second and third chapters, restricts attention to
bounded sets of the real line. The Lebesgue measure of a bounded set
and measurable functions are studied in Chapter 2 which finds its
highest point in the theorem of Egorov, of importance in establishing
convergence properties of integrals. The main elements of the theory of
Lebesgue integral are presented in Chapter 3, only for functions over
bounded sets, although the convergence theorems are proved establishing
the passage to the limit under the integral sign. Chapter 4 presents
the Lebesgue’s theorem about differentiability of monotone functions
and his versions of the fundamental theorems of calculus. The book
proposes 187 exercises where almost always the reader is proposed to
prove a statement. This reviewer thinks that this book is a very
helpful tool to get into Lebesgue’s theory in an easy manner.
rv: Daniel Cárdenas-Morales (Jaén)