This innovative textbook bridges the gap between undergraduate
analysis and graduate measure theory by guiding students from the
classical foundations of analysis to more modern topics like metric
spaces and Lebesgue integration. Designed for a two-semester
introduction to real analysis, the text gives special attention to
metric spaces and topology to familiarize students with the level
of abstraction and mathematical rigor needed for graduate study in
real analysis. Fitting in between analysis textbooks that are too
formal or too casual, From Classical to Modern Analysis is a
comprehensive, yet straightforward, resource for studying real
analysis. To build the foundational elements of real analysis, the
first seven chapters cover number systems, convergence of sequences
and series, as well as more advanced topics like superior and
inferior limits, convergence of functions, and metric spaces.
Chapters 8 through 12 explore topology in and continuity on metric
spaces and introduce the Lebesgue integrals. The last chapters are
largely independent and discuss various applications of the
Lebesgue integral. Instructors who want to demonstrate the uses of
measure theory and explore its advanced applications with their
undergraduate students will find this textbook an invaluable
resource. Advanced single-variable calculus and a familiarity with
reading and writing mathematical proofs are all readers will need
to follow the text. Graduate students can also use this
self-contained and comprehensive introduction to real analysis for
self-study and review.
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