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Unformatted text preview: Mathematics 105, Spring 2004 — M. Christ Final Exam Review Guide The final exam will primarily emphasize the portion of the course concerned with Lebesgue integration, in which we followed Stroock’s Chapters 2, 3, § 4.1 and §§ 5.0,2,3. From the first part of the course, you should know definitions, statements of theorems in- cluding hypotheses, and examples. You might be asked to state or to apply theorems, but you will not need to know details of proofs or of solutions to the problem sets. In format the final exam will be similar to the midterms. There will be short answer questions concerning definitions and examples; some of the latter may be in True-False format as in the second midterm exam. There will also be (short answer) questions about statements of theorems. Longer questions may ask you to reproduce portions of proofs of theorems and lemmas, to reproduce solutions of homework problems, or to solve new problems. I will aim for this exam to be approximately twice as long as the midterm exams; you’ll have three times as much time. Definitions. Compact sets. Open and closed sets. Differentiable functions, Df ( a ). Par- tial derivatives, Jacobian matrix. Directional derivatives, and their connection with Df . Second-order partial derivatives, continuously differentiable functions. σ-algebra, measure, measure space, measurable space. Probability measures, finite mea- sures, σ-finite measures. Outer measure, Lebesgue measurable sets, Lebesgue measure. F σ and G δ sets. Countable additivity, countable subadditivity. Complete measures, completion of a measure space. Borel sets, the Borel σ-algebra associated to any metric space. The σ-algebra σ ( E ; C ) generated by a collection C of subsets of E . π-systems and λ-systems. Axiom of choice. Characteristic (or indicator) functions, simple functions. Measurable functions, alter- native characterizations of measurability. Arithmetic in the extended real numbers....
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- Spring '04
- Math, measure, Lebesgue measure, Lebesgue integration, Aj Aj