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Unformatted text preview: THE GENERAL LEBESGUE INTEGRAL by Leonard R. Gardner III Submitted in Partial Fulfillment of the Requirements for Graduation with Honors from the South Carolina Honors College April 2000 Approved: Dr. Maria K. Girardi First Reader Dr. James W. Roberts Second Reader Peter C. Sederberg, James L. Stiver, or Douglas F. Williams for South Carolina Honors College Contents 1. Thesis Summary ii 2. Introduction iv 3. Measurable Functions 1 4. Measures 16 5. The Integral 29 6. Integrable Functions 38 7. The Lebesgue Spaces L p 46 8. Modes of Convergence 58 9. Conclusion 65 References 66 i 1. Thesis Summary This paper expands upon Robert Bartle’s exploration of the General Lebesgue Integral in his text Elements of Integration . Following Bartle’s example, the paper opens with a discussion of the groundwork on which the theory of the Lebesgue Integral stands. As a house sits on cement and brick footings, the Lebesgue Integral is propped on solid mathematical concepts such as algebras and measures. After setting the basics, the framework or the skeleton of the Integral is presented. Mathematicians are fond of handling the simple cases first then extending the results to more complicated and detailed cases. Obeying this methodology, the paper initially establishes the Integral for a limited class of functions, namely those measurable functions with nonnegative values. Then later the Integral is defined for all measurable functions. Having completed the base and the frame of the Integral, the paper then focuses on some of the beautiful and elegant theories that adorn the Gen eral Lebesgue Integral, like the Monotone Convergence Theorem and the Lebesgue Dominated Convergence Theorem. Indeed, convergence is an ex tremely important idea in mathematics and many times mathematicians are interested in sequences of functions and the convergence (if it does in fact converge) of these sequences.. Sometimes these sequences of functions con verge or get very close to another function. And other times these sequences diverge or don’t get very close to another function. In a convergent sequence of functions, no matter how far you go out in the sequence it still stays very close to one unchanging function or what mathematicians call the limit of the sequence. In a divergent sequence, as you move through the sequence you never get close to one particular function. What mathematicians want to know is if you take the limit of the sequence and then integrate will you get the same value as if you integrate each function in the sequence and then take the limit of the integrals? In other words, with two operations, does ii it matter in which order you perform them? It turns out that with some precautions, the order may be switched....
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This note was uploaded on 12/13/2011 for the course MATH 554 taught by Professor Girardi during the Fall '10 term at South Carolina.
 Fall '10
 Girardi

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