FOUNDATION OF ANALYSIS, LECTURE NOTES, FALL 2011 PART I: RIEMANN AND RIEMANN–STIELTJES INTEGRATION It may be a comfort to students just beginning to work in the calculus to know that Newton and Leibniz, two of the greatest mathematicians, did not fully under- stand what they themselves had produced M. Kline, Calculus 1. THE RIEMANNINTEGRAL 1.1. Definition. We consider a fixed bounded interval [a,b] and a bounded (real-valued) function f on [a,b]. A finite set C of points of [a,b] containing a and b is called a partition of [a,b]. Let C {c ,c ,...,c } be a partition of [a,b], where a c c c b, then Æ 0 1 n Æ 0 Ç 1 Ç ¢¢¢ Ç n Æ C max{c j c j 1, j 1,...,n} k k Æ ¡ ¡ Æ is called the mesh size of C . We say that ´ {´1,...,´n} is a choice sequence for C if c j 1 ´ j c j for Æ ¡ · · j 1,...,n. Given a partition C of [a,b] and a choice sequence ´, we define the Riemann sum of f Æ over C relative to ´ by n X R(f ;C ,´) f (´ j )(c j c j 1). Æ j 1 ¡ ¡ Æ Definition. We say that f is Riemann integrable on [a,b] if there is a number A R such that for any 2 " 0 there exists a ± 0 for which R(f ;C ,´) A " whenever C ±. È È j ¡ j Ç k k Ç Clearly if such number A exists then it is unique. Q1: Explain that! When f is Riemann integrable on [a,b], we call the corresponding number A the Riemann integral of f on [a,b] and write Z b Z A f (x)dx f (x)dx. Æ a Æ [a,b] In other words, we say that f is Riemann integrable if lim C 0 R(f ;C ,´) exists. Here the limit is not k k! standard, the precise definition is given above (there is A such that for any " 0 there exists ± etc). We È will later use limits like limf (®) a F (®) without additional explanations. ! The question when a function is Riemann integrable is not trivial, but it is not difficult to see (and we will do it in Section 3.1) that each continuous function is Riemann integrable. The complete descrip- tion of the Riemann integrable functions will be given later in the course. 1.2. Riemann–Darboux sums. Assume that m f (x) M on [a,b], and let C {c ,c ,...,c } be a · · Æ 0 1 n partition of [a,b]. We define m j inf{f (x): c j 1 x c j } and M j sup{f (x): c j 1 x c j }, Æ ¡ · · Æ ¡ · · and introduce the lower Riemann–Darboux sum of f over C : n X L (f ;C ) m j (c j c j 1), Æ j 1 ¡ ¡ Æ 1 2 RIEMANN AND RIEMANN–STIELTJES INTEGRATION and the upper Riemann–Darboux sum of f over C : n X U (f ;C ) M j (c j c j 1). Æ j 1 ¡ ¡ Æ Clearly, we have L (f ;C ) R(f ;C ,´) U (f ;C ) · · for any partition C and any choice sequence ´. Moreover, for any partition C , L (f ;C ) infR(f ;C ,´) and U (f ;C ) supR(f ;C ,´). Æ ´ Æ ´ 1.3. Criterion for Riemann integrability. We shall use the Riemann–Darboux sums to give a criterion for integrability that is easier to verify than the initial definition. Lemma 1.1. The function f is Riemann integrable if and only if lim L (f ;C ) lim U (f ;C ). C 0 Æ C 0 k k! k k! By this, we mean that limits above exist and are equal. Proof. Clearly, if f is Riemann integrable and A R b f (x)dx, then Æ a lim L (f ;C ) lim U (f ;C ) A. C 0 Æ C 0 Æ k k! k k! Q2: Write it down! Now suppose that lim C 0 L (f ;C ) lim C 0 U (f ;C ) A. It means that for any " 0 there exist ±1 k k! Æ k k! Æ È and ± such that L (f ;C ) A " whenever C ± and U (f ;C ) A " whenever C ± . We 2 j ¡ j Ç k k Ç 1 j ¡ j Ç k k Ç 2 let ± min(± ,± ), then Æ 1 2 R(f ;C ,´) A max( L (f ;C ) A , U (f ;C ) A ) " j ¡ j · j ¡ j j ¡ j · whenever C ±. Thus f is Riemann integrable on [a,b]. k k Ç Given two partitions C1 and C2 of [a,b], we say that C2 is more refined than C1 (or C2 is a refinement of C ) if C C , i.e., each point of C is a point of C . 1 1 ⊆ 2 1 2 Lemma 1.2. Suppose that C C are two partitions of [a,b]. 1 ⊆ 2 Then L (f ;C ) L (f ;C ) and U (f ;C ) U (f ;C ) for any bounded function f . 1 · 2 1 ¸ 2 Proof. We may add points to C one by one to get C . So it suffices to show that L (f ;C ) L (f ;C ) 1 2 1 · 2 where C C {c}. The inequality for the upper sums can be proved similarly, or one can use the 2 Æ 1 [ identity U (f ,C ) L ( f ,C ) together with the result for lower sums. Suppose that C {c ,c ,...,c } Æ¡ ¡ 1 Æ 0 1 n and c j 1 c c j . Then we have ¡ Ç Ç L (f ;C ) L (f ;C ) 2 ¡ 1 inf f (x)(c c j 1) inf f (x)(c j c) inf f (x)(c j c j 1) Æ c j 1 x c ¡ ¡ Å c x c j ¡ ¡ c j 1 x c j ¡ ¡ ¡ · · · · ¡ · · ³ ´ ³ ´ inf f (x) inf f (x) (c c j 1) inf f (x) inf f (x) (c j c) 0. Æ c j 1 x c ¡ c j 1 x c j ¡ ¡ Å c x c j ¡ c j 1 x c j ¡ ¸ ¡ · · ¡ · · · · ¡ · · It follows from the proof that L (f ;C ) L (f ;C ) (M m) C 2 ¡ 1 · ¡ k 1k if C2 is obtained from C1 by adding a single point. RIEMANN AND RIEMANN–STIELTJES INTEGRATION 3 Lemma 1.3. For any two partitions C1 and C2 of [a,b] L (f ;C ) U (f ;C ). 1 · 2 Proof. Let C C C , clearly, C is a partition of [a,b] which is more refined than both C and C . Æ 1 [ 2 1 2 Applying the previous lemma, we obtain L (f ;C ) L (f ;C ) U (f ;C ) U (f ;C ). 1 · · · 2 Lemma 1.4. The limits lim C 0 L (f ;C ) and lim C 0 U (f ;C ) exist and k k! k k! lim L (f ;C ) supL (f ;C ), lim U (f ;C ) infU (f ;C ). C 0 Æ C 0 Æ C k k! C k k! Proof. We shall prove that lim C 0 L (f ;C ) s, where s supC L (f ;C ). The second identity can be k k! Æ Æ proved in the same way. It is sufficient to show that for each " 0 there exists a ± 0 such that È È s L (f ;C ) " ¡ Ç whenever C ±. (The expression above is clearly positive, so we do not take the absolute value.) k k Ç By the definition of the supremum, there exists a partition C ¤ {c¤,...,c¤} such that L (f ;C ¤) Æ 0 n È s "/2. Now let C be any partition of [a,b], we write ¡ ¡ ¢ (1) L (f ;C ) L (f ,C C ¤) L (f ;C C ¤) L (f ;C ) . Æ [ ¡ [ ¡ Applying lemma 1.3, we have " (2) L (f ,C C ¤) L (f ,C ¤) s . [ ¸ È ¡ 2 Now, we want estimate L (f ;C C ¤) L (f ;C ), note that the partition C C ¤ is a refinement of C [ ¡ [ and it is obtained by adding less than n new points (n is the number of points in C ¤). By adding each new point to C we increase the lower sum by at most (M m) C , where M sup f (x) and ¡ k k Æ [a,b] m inf f (x). Thus, we get Æ [a,b] L (f ;C C ¤) L (f ;C ) (M m)n C . [ ¡ · ¡ k k Finally, we combine the last inequality with (1) and (2), " L (f ;C ) s (M m)n C s ", ¸ ¡ 2 ¡ ¡ k k È ¡ when C is small enough. k k We begin with ", find a partition C ¤ as above and let n n(") be the number of intervals in that Æ partition, finally we take " ± . Æ 2n(M m) ¡ Theorem 1.5. Let f be a bounded function on [a,b]. Then f is Riemann integrable if and only if sup L (f ;C ) inf U (f ;C ). C Æ C Proof. The theorem follows from Lemma 1.1 and Lemma 1.4. Corollary. Let f be a bounded function on [a,b]. Then f is Riemann integrable if and only if for any " 0 there is a partition C of [a,b] such that U (f ;C ) L (f ,C ) ". È ¡ Ç Q3: Explain how the statement above follows from the theorem. 4 RIEMANN AND RIEMANN–STIELTJES INTEGRATION 1.4. Riemann’s criterion for integrability. Assume that m f (x) M is a bounded function on [a,b].