Measure and Integration

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Measure

   S. Kesavan

   Abstract

   This chapter studies the basic properties of abstract measures defined on classes of subsets of a non-empty set, like rings, σ-rings and σ-algebras. The method of Carathéodory of completing a measure defined on a ring, using outer-measures, is described. The method of completing a measure by adding sets of measure zero is also discussed.

3. Chapter 2. The Lebesgue measure

   S. Kesavan

   Abstract

   It is shown that the notion of the length of an interval defines a measure on the ring generated by left-closed-right-open intervals in the real line ℝ. Using the method of Carathéodory, the Lebesgue measure is constructed on ℝ and its important properties are studied. Generalizations to ℝ^N,N ≥ 2, are also described. The property of translation invariance, and its consequences, are discussed.

4. Chapter 3. Measurable functions

   S. Kesavan

   Abstract

   The notion of measurability of real-valued functions defined on a set endowed with a σ-algebra is introduced and important properties of measurable functions are studied. The construction of the Cantor function, which is a rich source of counter-examples, is described.

5. Chapter 4. Convergence

   S. Kesavan

   Abstract

   Three notions of convergence of sequences of measurable functions, viz. pointwise convergence almost everywhere, almost uniform convergence, and convergence in measure, are introduced and their inter-relationships are examined. It is also shown that each type of convergence is also equivalent to the corresponding Cauchy criterion.

6. Chapter 5. Integration

   S. Kesavan

   Abstract

   The Lebesgue integral is defined in stages: first for non-negative simple functions, then for non-negative measurable functions and, finally, for a class of functions which are `integrable’. Important limit theorems are proved. The real line has now two notions of integration, repectively, those of Riemann and Lebesgue. Their relationship is examined. A measure-theoretic proof of the Weierstrass approximation theorem is presented. A glossary of terms from probability thoery is given and they are explained in terms of notions from measure theory.

7. Chapter 6. Differentiation

   S. Kesavan

   Abstract

   The fundamental theorem of classical calculus shows that differentiation and integration are really two sides of the same coin. In this chapter, various classes of functions which are differentiable almost everywhere are studied and the condition for a function, which is differentiable almost everywhere with the derivative being integrable, being expressed as the indefinite integral of the derivative, is examined.

8. Chapter 7. Change of variable

   S. Kesavan

   Abstract

   The notion of Fréchet differentiability in ℝ^N is introduced. The effect of the action of a diffeomorphism on the Lebesgue measure and Lebesgue integrals are studied.

9. Chapter 8. Product Spaces

   S. Kesavan

   Abstract

   Given two σ-finite measure spaces, the product σ-algebra and the product measure is defined on the product space. Fubini’s theorem, which gives conditions under which the integral of a function on the product space can be expressed as an iterated integral of successive integration on the original spaces, is proved. Several consequences of this result are presented through examples. The integration of a radial function, using polar coordinates in ℝ^N is studied.

10. Chapter 9. Signed measures

    S. Kesavan

    Abstract

    A signed measure on a measurable space is a set function which has all the properties of a measure, except that of non-negativity. It is shown that signed measures are essentially got by taking the difference of two measures. The notion of absolute continuity is introduces and the famous Radon-Nikodym theorem is proved for σ-finite signed measures. The notion of singularity, of one measure with respect to another, is studied.

11. Chapter 10. Lp-spaces

    S. Kesavan

    Abstract

    This chapter is devoted to the study of the Lebesgue spaces, also known as the L^p-spaces, where 1 ≤ p ≤ ∞. It is shown that these are Banach spaces and their duals are identified. Important properties of the convolution product are studied.

12. Backmatter