A Course in Analysis : Vol. III

In 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 L^p-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 L^p-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.