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Useful both as a text for students and as a source of reference for the more advanced mathematician, this book presents a unified treatment of that part of measure theory which is most useful for its application in modern analysis. Topics studied include sets and classes, measures and outer measures, measurable functions, integration, general set functions, product spaces, transformations, probability, locally compact spaces, Haar measure and measure and topology in groups. The text is suitable for the beginning graduate student as well as the advanced undergraduate.
My main purpose in this book is to present a unified treatment of
that part of measure theory which in recent years has shown itself
to be most useful for its applications in modern analysis. If I
have accomplished my purpose, then the book should be found usable
both as a text for students and as a sour ce of refer ence for the
more advanced mathematician. I have tried to keep to a minimum the
amount of new and unusual terminology and notation. In the few pI
aces where my nomenclature differs from that in the existing
literature of meas ure theory, I was motivated by an attempt to
harmonize with the usage of other parts of mathematics. There are,
for instance, sound algebraic reasons for using the terms "lattice"
and "ring" for certain classes of sets-reasons which are more
cogent than the similarities that caused Hausdorff to use "ring"
and "field. " The only necessary prerequisite for an intelligent
reading of the first seven chapters of this book is what is known
in the Uni ted States as undergraduate algebra and analysis. For
the convenience of the reader, 0 is devoted to a detailed listing
of exactly what knowledge is assumed in the various chapters."
As a newly minted Ph.D., Paul Halmos came to the Institute for
Advanced Study in 1938--even though he did not have a
fellowship--to study among the many giants of mathematics who had
recently joined the faculty. He eventually became John von
Neumann's research assistant, and it was one of von Neumann's
inspiring lectures that spurred Halmos to write "Finite Dimensional
Vector Spaces." The book brought him instant fame as an expositor
of mathematics.
"Finite Dimensional Vector Spaces" combines algebra and geometry
to discuss the three-dimensional area where vectors can be plotted.
The book broke ground as the first formal introduction to linear
algebra, a branch of modern mathematics that studies vectors and
vector spaces. The book continues to exert its influence sixty
years after publication, as linear algebra is now widely used, not
only in mathematics but also in the natural and social sciences,
for studying such subjects as weather problems, traffic flow,
electronic circuits, and population genetics.
In 1983 Halmos received the coveted Steele Prize for exposition
from the American Mathematical Society for "his many graduate texts
in mathematics dealing with finite dimensional vector spaces,
measure theory, ergodic theory, and Hilbert space."
2014 Reprint of 1962 Edition. Full facsimile of the original
edition, not reproduced with Optical Recognition Software. In
"Algebraic Logic" Halmos devised polyadic algebras, an algebraic
version of first-order logic differing from the better known
cylindric algebras of Alfred Tarski and his students. An elementary
version of polyadic algebra is described in monadic Boolean
algebra. This book addresses some of the problems of mathematical
logic and the theory of polyadic Boolean algebras in particular. It
is intended to be an efficient way of treating algebraic logic in a
unified manner.
2013 Reprint of 1963 Edition. Full facsimile of the original
edition, not reproduced with Optical Recognition Software. The
theory of Boolean algebras is one of the most attractive parts of
mathematics. On the one hand, Boolean algebras arise naturally in
such diverse fields as logic, measure theory, topology, and ring
theory, so that the study of these objects is motivated by
important applications. At the same time, the theory which has been
developed constitutes one of the most elegant parts of modern
algebra. Finally, the subject still poses many challenging
questions, some of which have considerable importance. A graduate
student who wishes to study Boolean algebras will find several
excellent books to smooth his way: for an introduction, the book by
Halmos is probably the best of these monographs. It offers a quick
route to the most attractive parts of the theory.
2013 Reprint of 1956 Edition. Full facsimile of the original
edition, not reproduced with Optical Recognition Software. Ergodic
theory is a branch of mathematics that studies dynamical systems
with an invariant measure and related problems. Its initial
development was motivated by problems of statistical physics. A
central concern of ergodic theory is the behavior of a dynamical
system when it is allowed to run for a long time. Paul Richard
Halmos (1916 - 2006) was a Hungarian-born American mathematician
who made fundamental advances in the areas of probability theory,
statistics, operator theory, ergodic theory, and functional
analysis (in particular, Hilbert spaces). He was also recognized as
a great mathematical expositor.
2012 Reprint of 1942 Edition. Exact facsimile of the original
edition, not reproduced with Optical Recognition Software. As a
newly minted Ph.D., Paul Halmos came to the Institute for Advanced
Study in 1938--even though he did not have a fellowship--to study
among the many giants of mathematics who had recently joined the
faculty. He eventually became John von Neumann's research
assistant, and it was one of von Neumann's inspiring lectures that
spurred Halmos to write "Finite Dimensional Vector Spaces." The
book brought him instant fame as an expositor of mathematics.
Finite Dimensional Vector Spaces combines algebra and geometry to
discuss the three-dimensional area where vectors can be plotted.
The book broke ground as the first formal introduction to linear
algebra, a branch of modern mathematics that studies vectors and
vector spaces. The book continues to exert its influence sixty
years after publication, as linear algebra is now widely used, not
only in mathematics but also in the natural and social sciences,
for studying such subjects as weather problems, traffic flow,
electronic circuits, and population genetics. In 1983 Halmos
received the coveted Steele Prize for exposition from the American
Mathematical Society for "his many graduate texts in mathematics
dealing with finite dimensional vector spaces, measure theory,
ergodic theory, and Hilbert space."
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