Libraries and resellers, please contact cust-serv ams. See our librarian page for additional eBook ordering options. Integration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been interpreted in terms of measure theory. This introductory text starts with the historical development of the notion of the integral and a review of the Riemann integral. From here, the reader is naturally led to the consideration of the Lebesgue integral, where abstract integration is developed via measure theory.

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The proof is an application of the Radon-Nikodym theorem. An independent proof of the Radon-Nikodym theorem was also given in theorem Before we prove this, we observe the following. Proof: The implication i ii is obvious. Conversely, suppose T is continuous but i does not hold. On the other hand, I 0 This suggests an application of the Radon-Nikodym theorem for v to locate g. Then v is a well-defined finite signed measure, and v Step 2.

Step 3. We show that this implies the required claim of step 2, i. Assume first that 1 i of nonnegative simple measurable functions increasing to IgIq Let. Since this holds b n, by the monotone convergence theorem, we have I: J Using theorem 8. Further, from proposition Remarks: i The claim of proposition Note that S is a nonempty set.

Ti-oo Let 00 W. These are called decomposable measure spaces. For this, we refer the reader to Hewitt and Stromberg [18]. Appendix A Extended real numbers A. Note that lien inf xn. Let a : N -- N be Let any bijective map. Similarly, for a finite family of nonempty sets Al , We can think of Al x However, this cannot be proved with the usual axioms of set theory.

For a short introduction to axiomatic set theory, see Rana [30]. For detailed account of axiomatic set theory, the axiom of choice, and its history, see Halmos [15] and Fraenkel [12]. The axiom of choice finds applications in many diverse branches of mathematics. We have used it in section 4. We say X and Y are equipotent if there exists a bijection between them. We write this as X ti Y. A set which is not countable is called uncountable.

For example, N, Z, Q are all countable sets while R is uncountable. For a detailed discussion, see Rana [30]. Let C be a collection of sets such that any two members of C are equipotent to each other. Then one can assign a symbol, called its cardinal number, to each A E C, denoted by card A.

The cardinal number of a set A is also called the cardinality of A. Can the same be said about arbitrary sets which are not necessarily finite?

We write card A Is card A This raises the following natural question:. Does there exist a cardinal number a such that? The answer to this question is not known. An equivalent formulation of this is the following: The set JR can be well-ordered in such a way that each element of JR is preceded by only countably many elements.

We used this in section 3. It is known that the continuum hypothesis is independent of the Zermelo-Fraenkel axioms of set theory. Appendix D Urysohn's lemma In theorem 8. This follows from the following result, known as Urysohn's lemma: D. Proof: This follows from theorem D. Corollary: Let X, d be a metric space and x, y E X, x 4 y.

The second corollary has a great significance. It ensures that metric spaces have a rich supply of real-valued continuous functions: any two distinct points can be separated by a real-valued continuous function. Urysohn's lemma can be extended to some topological spaces. For details see Hewitt and Stromberg [18], Munkres [26].

Appendix E Singular value decomposition of a matrix We consider matrices with real entries only. Theorem: Let A be an m x n real matrix of rank r.

Since B is symmetric, all its eigenvalues are real. In fact, it is easy to see that all the eigenvalues of B are nonnegative and there exists a complete set of orthonormal eigenvectors of B.

A, Let xl, Let P be the m x r matrix whose column vectors are 4, Corollary: Let A be an n x n real nonsingular matrix. Note: The positive real numbers X1, VA-r, obtained in the proof of theorem E.

The extended real number ab f is called the total variation of f over [a, b]. The function f is said to be of bounded variation if ab f F. Example: Suppose f R is a monotonically increasing or [a, b] monotonically decreasing function. Proposition: For functions f and g on [a, b], the following hold: i If f is of bounded variation, then f is a bounded function.

Proof: Exercise. Proposition: Let f [a, b] ][8 be of bounded variation. This proves iv. Theorem Jordan : Let f [a, b] R. Then f is of bounded variation iff f is the difference of two monotonically increasing functions. Conversely, let f be of bounded variation. Then g and h are monotonically increasing, by proposition F. Exercise: i Let f : [a, b] R be a differentiable function such that its derivative is a bounded function. Show that f has bounded variation.

Hint: Use Lagrange's mean value theorem. Show that f is a continuous function of bounded variation. Show that f is uniformly continuous but is not of bounded variation.

Then the , functions T2,1 Appendix G ][8'n be differentiable at a c U. Theorem: Let U be an open subset of ][8n and T : U -- ][8m be a mapping with coordinate functions Tl, Then the following hold: i T is differentiable at a E U iff each Ti : U - ][8 is differentiable at a. Further, in that case the itl' row of the matrix of dT a is the matrix vector of dT2 a. Let Ai denote the ith coordinate map of the linear map dT a.

Thus the ith row in the matrix of dT a is the 1 x n matrix of dT a. Conversely, suppose each Ti is differentiable at a. Consider the linear map A whose ith coordinate function is dTi a.

This proves i. Appendix G We show that dT a is indeed as defined above. We prove it by induction on n. So assume this for n - 1. Let the matrix of dT a with respect to a basis of Rn be denoted by [ dT a ], i. The Jacobian of T at a is defined to be the determinant of the matrix [ dT a ] and is denoted by JT a.

Proof S. Kumaresan : Since T is a Cl-map, by theorem G. Let a E U be fixed arbitrarily. This proves W. We construct a recursively as follows.


An Introduction to Measure and Integration: Second Edition

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An Introduction to Measure and Integration

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