**Integration for Dirichlet function (10 pts) Warning: If you are still not familiar**

Bonus Problems Due June 15th, before the final exam You are allowed to use any material as you want, to search the internet, to discuss with any persons for the problems before answering the questions. Write down the answer by yourself, and try to present all you have learnt from reading. Organize your words properly. No credit will be given for ambiguous answers. 1 Integration for Dirichlet function (10 pts) Warning: If you are still not familiar with the notation of Riemannian integral, please do not start. A good way to check you have mastered the knowledge of Riemann integral is that you can finish all the problems of 5.1-5.4 on our textbook. As we have learnt in class, the Dirichlet function D(x) = ( 1, x ∈ Q 0, x 6∈ Q is not Riemannian integrable on [0, 1]. Can you define another notation of integration properly, such that the integral of Dirichlet function is meaningful? Hint: Read chapter 1, sections 1-4, and chapter 2 sections 1 and 2 of the book ‘Real analysis: measure theory, integration and Hilbert spaces, by Elias Stein & Rami Shakarchi, (2005), Princeton University Press’. Answer the following questions. 1. What is exterior measure? Five observations of exterior measure. (section 1.2) 2. What is Lebesgue measurable set? Properties? (section 1.3) 3. What is σ-algebra? What is Borel set? 4. Construct a non-measuable set in R. 5. What is measurable functions? Properties? Approximation of measurable functions. (section 1.4) 6. Littlewood’s three principles. 1 Figure 1: Upper: Riemannian integral. Lower: Lebesgue integral. 7. How to define Lebesgue integral? Four stages. 8. Space L 1 . What is the norm and the metric of L 1 ? 9. Calculate the Lebesgue integral of D(x) on [0, 1]. 2 Change of variables (10 pts) Study the change-of-variables theorem by yourself and solve the following problems. (2 points) Solve the following improper integral Z ∞ −∞ e −x 2 dx. (3 points) Find the area bounded by the curve ( x 2 a 2 + y 2 b 2 ) 2 = xy c 2 , (a, b, c > 0). (5 points) Find the volume of the unit ball in R n . 2

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