Real Analysis is the third volume in the Princeton Lectures in
Analysis, a series of four textbooks that aim to present, in an
integrated manner, the core areas of analysis. Here the focus is on the
development of measure and integration theory, differentiation and
integration, Hilbert spaces, and Hausdorff measure and fractals. This
book reflects the objective of the series as a whole: to make plain the
organic unity that exists between the various parts of the subject, and
to illustrate the wide applicability of ideas of analysis to other
fields of mathematics and science.
After setting forth the basic facts of measure theory, Lebesgue
integration, and differentiation on Euclidian spaces, the authors move
to the elements of Hilbert space, via the L2 theory. They next present
basic illustrations of these concepts from Fourier analysis, partial
differential equations, and complex analysis. The final part of the book
introduces the reader to the fascinating subject of
fractional-dimensional sets, including Hausdorff measure,
self-replicating sets, space-filling curves, and Besicovitch sets. Each
chapter has a series of exercises, from the relatively easy to the more
complex, that are tied directly to the text. A substantial number of
hints encourage the reader to take on even the more challenging
exercises.
As with the other volumes in the series, Real Analysis is accessible
to students interested in such diverse disciplines as mathematics,
physics, engineering, and finance, at both the undergraduate and
graduate levels.
Also available, the first two volumes in the Princeton Lectures in
Analysis:
