'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.
