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Unformatted text preview: Notes on integration and measure P. B. Kronheimer, for Math 114 Revised, September 12, 2010 These are some outline notes on integration and measure, based on the book by Stein and Sharkarchi that we are using in class, but adapted to the approach we will be following. These notes may be helpful in conjunction with the book, for the topics which we will be covering at the start of the semester; but taken by themselves they won’t be very useful, and they lack detail, examples, and explanation. Please read the book too. It is excellent. 1. Introduction Integrals and measure The volume of the unit ball in 3-space is 4 π/ 3. This is a familiar statement, often proved in the second semester of a standard calculus course, using integration. We have no trouble, in this particular case, grasping the intuitive idea of “volume”. But the concept is a di ffi cult one mathematically. We might believe that we can assign to any bounded subset of R 3 a notion of its “vol- ume” (or total mass), and that this notion of volume should have some basic properties such as these: • if a set E ⊂ R 3 is decomposed into two disjoint pieces, E 1 and E 2 , then the volume of E should be the sum of the volumes of E 1 and E 2 ; • if E is obtained from E by a rigid motion (rotation or translation), then E and E should have the same volume; • the volume of a unit ball (for example) should be 4 π/ 3. 2 2. Measurable sets: Lebesgue measure We would like to have a mathematical definition of such a notion of volume, that fits our intuitive idea: build a copy of E out of fine sand; put the sand in a pan and weigh it; divide the weight by the density of the sand. Unfortunately there is no general notion of the “volume” of an arbitrary subset of R 3 which enjoys the properties just listed. This is revealed most sharply by the “Banach- Tarski paradox”. It is possible to decompose a unit ball into finitely many disjoint pieces and reassemble these pieces (using rigid motions) to form two balls, each identical to the original. The shapes of these pieces are so pathological that we cannot model them out of our “sand”. No sand is fine enough. Indeed, the very existence of these pieces can only be established using the Axiom of Choice (which means that we cannot give a completely explicit description of them). While the word “volume” is appropriate in dimension 3, we use the generic word “measure” when we work in R d . So “measure” has the meaning of “area” when d = 2, and “length” when d = 1. The notations of area, volume and measure are closely tied to the study of integration (for we may try to define the integral as the signed “area under the graph”). The Banach-Tarski paradox reveals that mathematics allows for some very strange sets and very strange functions. We should not expect every set to be measurable, and we should not expect every function to be integrable (even bounded functions on [0 , 1]). It was Lebesgue’s achievement to correctly isolate the type of subsets in1])....
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This note was uploaded on 08/30/2011 for the course MATH 114 taught by Professor Youngmoney during the Spring '10 term at Adelphi.
- Spring '10