A Concise Introduction to Measure Theory, Satish Shirali

 A Concise Introduction to Measure Theory, Satish Shirali

      Chap. 1 starts with. For other kinds of functions, it is not always clear that a distribution function exists at all. A countably additive measure as the “size” of a set is then presented as motivated by the need for a limit of functions having distributions again to have a distribution.

        The concept of a countably additive measure on a r-algebra and measurability of functions are introduced in Chap. 2 against the backdrop of the motivation described earlier. The integral of a nonnegative measurable function is defined as the improper integral of its distribution function, and the monotone convergence theorem is proved. The integral of a function that may take negative values is introduced, but its properties are dealt with in the next chapter.

       Simple functions are defined in Chap. 3 and used to establish the additivity of the integral. The dominated convergence theorem, for which the additivity of the integral is used, is also treated in this chapter. There is a discussion of the extension to subadditive fuzzy measures, but this material is optional.

       Chapter 4 is about constructing Lebesgue measure. After defining Lebesgue outer measure, Carathéodory’s ideas are applied to an abstract outer measure in the usual manner. It is shown that the classical Riemann integral agrees with the Lebesgue integral for Riemann integrable functions, and the existence of a nonmeasurable subset of the reals is discussed. The chapter ends with induced measures, which find application in connection with product measures and in identifying certain improper integrals as being Lebesgue integrals.

        The counting measure and interchanging the order of summation in a repeated sum as amounting to interchanging the order of integration are treated at length in Chap. 5. In this connection, the unconditional sum is identified with the integral respect to the counting measure. 

       Product measures and the theorems of Tonelli and Fubini are taken up in Chap. 6. In Chap. 7, the relation to differentiation is discussed in some detail. There are many approaches to the topic, the one adopted in this book being via the Vitali covering theorem and Tonelli’s theorem. While the concept of total variation is

presented ad hoc in most texts, here it is presented as a natural outcome of attempts to decompose a function as the sum of an increasing and a decreasing function.

      The relation between differentiation and Lebesgue integration is not as straightforward as in the case of Riemann integration, and no discussion of the matter can be complete without the Cantor set and function. Their essentials are discussed in Chap. 8.

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