Chapter 1 Preliminary 行路难！行路难！多歧路，今安在？ 长风硫浪会有时直辈否膏露潜 1.1 Introduction This lecture note is prepared for the course Introduction to Real analysis and Fourier analysis.It can into two parts.T hrst part the tegatioi number of discontimous pointsis countable.Therefore the integr almost continuous functions.Even though the great triumph was achieved by the Riemannian stha Indeed continuous functions are convergent to f then 1.f may not be Riemannian integrable; 2.even f is Riemannian integrable, 。in(h=厂feld may not hold. We give a counter-example for item 1 in the above.We can enumerate all rational numbers in [0,1]as (a,q2.........define 1a)-{bm0…9 It follows that f converges to the Dirichlet function D(),which is not Riemannian integrable. 5 Chapter 1 Preliminary 1¥Jú1¥Júı‹¥ß8S3º ºªL¨kûßÜ!~LÙ°" ))ox51¥J6 1.1 Introduction This lecture note is prepared for the course Introduction to Real analysis and Fourier analysis. It can be roughly divided into two parts. The main subject in the first part is the Lebesgue’s integration theory. We have learned in Calculus that a function is Riemannian integrable if and only if the number of discontinuous points is countable. Therefore the Riemannian integral mainly works with almost continuous functions. Even though the great triumph was achieved by the Riemannian integral, it still has a major defect: not working well with limit. Indeed, continuous functions are not closed under taking limit, i.e., the limit of sequence of continuous functions is not necessarily continuous. Moreover, let fn be a sequence of Riemannian integrable functions on [0, 1], which is convergent to f then 1. f may not be Riemannian integrable; 2. even f is Riemannian integrable, limn→∞ Z 1 0 fn(x)dx = Z 1 0 f(x)dx may not hold. We give a counter-example for item 1 in the above. We can enumerate all rational numbers in [0, 1] as {q1, q2, · · · , ...}, define fn(x) =  1, x = q1, q2, · · · , qn; 0, else. It follows that fn converges to the Dirichlet function D(x), which is not Riemannian integrable. 5