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real - REVIEW OF LEBESGUE MEASURE AND INTEGRATION...

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REVIEW OF LEBESGUE MEASURE AND INTEGRATION CHRISTOPHER HEIL These notes will briefly review some basic concepts related to the theory of Lebesgue measure and the Lebesgue integral. We are not trying to give a complete development, but rather review the basic definitions and theorems with at most a sketch of the proof of some theorems. These notes follow the text Measure and Integral by R. L. Wheeden and A. Zygmund, Dekker, 1977, and full details and proofs can be found there. 1. OPEN, CLOSED, AND COMPACT SUBSETS OF EUCLIDEAN SPACE Notation 1.1. N = { 1 , 2 , 3 , . . . } is the set of natural numbers, Z = { . . . , 1 , 0 , 1 , . . . } is the set of integers, Q is the set of rational numbers, R is the set of real numbers, and C is the set of complex numbers. R d is real d -dimensional Euclidean space, the space of all vectors x = ( x 1 , . . . , x d ) with x 1 , . . . , x d R . On occasion, we formally use the extended real number line R ∪ {−∞ , ∞} = [ −∞ , ], but it is important to note that is a formal object, not a number. To write a [ −∞ , ] means that either a is a finite real number or a is one of ±∞ . We write | a | < to mean that a is a finite real number. Note that there is no analogue of the extended reals when we consider complex numbers; there’s no obvious “ ” or “ −∞ .” We declare some arithmetic conventions for the extended real numbers: + = , 1 / 0 = , 1 / = 0, and 0 · ∞ = 0. The symbols ∞ − ∞ are undefined, i.e., they have no meaning. The empty set is denoted by . Two sets A , B are disjoint if A B = . A collection { A k } of sets are disjoint if A j A k = whenever j negationslash = k . The real part of a complex number z = a + ib is Re ( z ) = a , and the imaginary part is Im ( z ) = b . The complex conjugate of z = a + ib is ¯ z = a ib . The modulus , or absolute value , of z = a + ib is | z | = z ¯ z = a 2 + b 2 . square For concreteness, we will use the Euclidean distance on R d in these notes. However, all the results of this section are valid with respect to any norm on R d . Definition 1.2. (a) The Euclidean norm on R d is | x | = ( x 2 1 + · · · + x 2 d ) 1 / 2 . The distance between x , y R d is | x y | . Date : Revised July 30, 2007. c circlecopyrt 2007 by Christopher Heil. 1
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2 REVIEW OF LEBESGUE MEASURE AND INTEGRATION (b) Suppose that { x n } n N is a sequence of points in R d and that x R d . We say that x n converges to x , and write x n x or x = lim n →∞ x n , if lim n →∞ | x n x | = 0 . square Using this definition of distance, we can now define open and closed sets in R d and state some of their basic properties. Definition 1.3. Let E R d be given. (a) E is open if for each point x E there is some open ball B r ( x ) = { y R d : | x y | < r } centered at x that is completely contained in E , i.e., B r ( x ) E for some r > 0. (b) A point x R d is a limit point of E if there exist points x n E that converge to x , i.e., such that x n x .
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