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Course In Analysis, A - Vol. Iii: Measure And Integration Theory, Complex-valued Functions Of A Complex Variable (Paperback)
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Course In Analysis, A - Vol. Iii: Measure And Integration Theory, Complex-valued Functions Of A Complex Variable (Paperback)
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'It is a great book for a first year (US) graduate student. One of
the nice features of the book is that the book contains full
solutions for all of the problems which make it useful as reference
for self-study or qualifying exam prep.' (See Full Review)MAA
ReviewsIn this third volume of 'A Course in Analysis', two topics
indispensible for every mathematician are treated: Measure and
Integration Theory; and Complex Function Theory.In the first part
measurable spaces and measure spaces are introduced and
Caratheodory's extension theorem is proved. This is followed by the
construction of the integral with respect to a measure, in
particular with respect to the Lebesgue measure in the Euclidean
space. The Radon-Nikodym theorem and the transformation theorem are
discussed and much care is taken to handle convergence theorems
with applications, as well as Lp-spaces.Integration on product
spaces and Fubini's theorem is a further topic as is the discussion
of the relation between the Lebesgue integral and the Riemann
integral. In addition to these standard topics we deal with the
Hausdorff measure, convolutions of functions and measures including
the Friedrichs mollifier, absolutely continuous functions and
functions of bounded variation. The fundamental theorem of calculus
is revisited, and we also look at Sard's theorem or the
Riesz-Kolmogorov theorem on pre-compact sets in Lp-spaces.The text
can serve as a companion to lectures, but it can also be used for
self-studying. This volume includes more than 275 problems solved
completely in detail which should help the student further.
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