2.1 Elementary sets and (pre)content.

Transcription

1 10 2. PEANO-JORDAN CONTENT The soewhat dissatisfactory situatio with the Riea itegral led Cator, Peao ad particularly Jorda develop a theory of (what we ow call) the Peao-Jorda cotet (a.k.a. just as Jorda cotet). The uderlyig otivatio is that, sice the Riea itegral is after all a eas to defie the area of a regio i R 2 aely that bouded betwee oe of the axes ad the graph of a fuctio we ay as well try to address the otio of area directly fro the outset. The usefuless of this philosophical tur of thought becoes eve ore apparet whe higher-diesioal Riea itegrals are cosidered; there oe ofte wats to itegrate f oly over a bouded regio i, say, R 2 or R 3 ad the eve itegratig fuctio f = 1 over such regios leads to proble with defiig areas ad volues. Copared to the Lebesgue easure, the Peao-Jorda cotet has uerous shortcoigs (essetially idetical to those of the Riea itegral) ad so, for preset day atheatics, its relevace rests ostly i the historical role it played at its tie. However, the developet of Peao-Jorda cotet will deostrate ay techical aspects of a full-fledged theory of easure ad so we ay start with it just as well. 2.1 Eleetary sets ad (pre)cotet. The developet of ay theory of cotet or easure has essetially two parts: (1) Idetify a class of eleetary sets for which we agree what their cotet/easure is. (2) Exted this to a larger class, called easurable sets, by way of approxiatios fro withi ad without by eleetary sets. Of course, there are uerous aspects oe should be cocered about i ipleetig such a progra. For istace, if the class of eleetary sets is chose too large, or the specific eaig of cotet we agree o is poorly behaved, the approxiatios cosidered i (2) ight lead to icosistet aswers eve for soe eleetary set. O the other had, if the class of eleetary sets is too sall, too few additioal sets could be reached by approxiatios. As we shall see later, these are exactly the cocers that took quite a while to tue out correctly. For the Peao-Jorda cotet, our choice of eleetary set will be as follows: Defiitio 2.1 (Eleetary sets) A half-ope box i R d is a set of the for I :=(a 1,b 1 ] (a 2,b 2 ] (a d,b d ] (2.1) where < a i < b i < for each i = 1,...,d. A eleetary set is the ay uio of the for S I i where I i 2 I for all i = 1,...,. We will write I to deote the set of all half-ope boxes icludig the epty set ad E for the class of eleetary sets. Lea 2.2 The class E is closed uder uios, itersectios ad set differeces. Proof. The closure uder uios is trivial. For itersectios we use that if I, J are o-disjoit half-ope boxes, the so is I \ J. For set differeces we use that if I,J are half-ope boxes, the J r I is either epty or a fiite uio of half-ope boxes. To exted this to eleetary sets, we just apply basic set operatios.

2 For a half-ope box I as i (2.1) we the defie its (pre)cotet by I := d (b i a i ). (2.2) with the proviso /0 = 0. Whe a set is a uio of a fiite uber of disjoit half-ope boxes {I i }, our ituitio would dictate to set its (pre)cotet to the su I i. However, this harbors a cosistecy proble: We do ot kow that a differet represetatio as a su of half-ope boxes would result i the sae value. This, ad a few other issues that are related to this proble, is addressed i: Propositio 2.3 (Cosistecy, subadditivity, additivity) Let {I i : i = 1,...,} I ad {J j : j = 1,...,} I with I i \ I i 0 = /0 wheever i 6= i 0. The [ [ I i J j ) I i apple J j. (2.3) I particular, if the uios are equal ad {J j : j = 1,...,} are disjoit, the equality holds. We begi with a eleetary versio of the clai: Lea 2.4 Let {I i : i = 1,...,} I be disjoit with I := S I i 2 I. The I = I i. (2.4) Proof i d = 1. Suppose I is a half-ope iterval ad {I i : i = 1,...,} are disjoit half-ope itervals such that I = S I i. We ca always relabel the itervals such that I i =(a i,b i ] for < a 1 < b 1 apple a 2 < b 2 apple applea < b <. (2.5) But the the fact that the uio of I i s ust be a iterval forces b i = a i+1 for all i = 1,..., 1 alog with a 1 = a ad b = b. This ad the specific for iplies I i = ad so we get I = I i as desired. (a i,b i ] =(b i a i ) (2.6) b i a i )= i a i b 1 = b + b i a i a = b a i=2 Proof i d 2. The proof has two steps. First we address the case of (what we will call) product partitios. The we will use that to hadle geeral partitios as well. STEP 1 (Product partitios): By a product partitio of a half-ope box I we will ea ay collectio of sets of the for J 1, j1 J 1, jd such that (1) J i, j is a half-ope iterval i R for all i = 1,...,d ad all j = 1,..., i, (2) J i, j \ J i,k = /0 uless j = k, ad 11 (2.7)

3 12 (3) their uio is all of I, i.e., I = 1 [ j 1 =1 d [ j d =1 (J 1, j1 J d, jd ). (2.8) Naturally, the uio i (2.11) is disjoit thaks to (2). Observe also that K i := i [ J i, j (2.9) is ecessarily a half-ope iterval ad I = K 1 K d. Give such a product partitio, we ca use the product for of (2.2) ad the distributive law for ultiplicatio aroud suatio to get 1 j 1 =1 d j d =1 J 1, j1 J d, jd = 1 j 1 =1 d j d =1 d J i, ji = d i J i, j (2.10) Fro the (already proved) clai i d = 1 we kow that i J i, j i = K i. Relabelig thigs we coclude that 1 j 1 =1 d j d =1 This verifies the clai i the case of product partitios. J 1, j1 J d, jd = I. (2.11) STEP 2 (Geeral partitios): Let I 2 I ad suppose that I = S I i for soe {I i } I with I i \ I j = /0 wheever i 6= j. Sice I i is itself a product of half-ope itervals, for each k = 1,...,d we ca euerate all edpoits of all itervals i the k-th coordiate directios that appear i the product costitutig I i for soe i. Labelig these edpoits icreasigly, for each k = 1,...,d we get a sequece a k,1 apple a k,2 apple applea k,2 1 apple a k,2. (2.12) We ca ow use these to defie a product partitio S J j of I, where =(2) d ad each J j is a half-ope box of the for (a 1,i1,a 1,i1 +1] (a d,id,a d,id +1] (2.13) (which is epty wheever a k,ik +1 = a k,ik for ay k = 1,...,d). The I = J j (2.14) by STEP 1. Thaks to disjoitess of {I i }, each J j has a o-epty itersectio with at ost oe I i, ad sice both {I i } ad {J j } are partitios of I, we have J j \ I i 6= /0 ) J j I i. (2.15) Settig S i := { j : J j \I i 6= /0 }, it the follows that {J j : j 2 S i } costitutes a product partitio of I i, for each i = 1,...,. By STEP 1 agai, we thus have j2s i J j = I i, i = 1,...,d, (2.16)

4 13 ad suig this over i with the help of (2.14) the yields I = J j = j2s i J j = I i (2.17) where we used that S i \ S i 0 = /0 whe i 6= i 0 ad S S i = {1,...,}. By sall odificatios of the proof, we also get: Corollary 2.5 Let {I i : i = 1,...,} I ad I 2 I. The [ I I i ) I apple I i. (2.18) Proof. Repeatig the arguet i STEP 2 of the previous proof, we ca write S I i as S J j where J j 2 I ad {J j } is a product partitio arisig fro collectig all edpoits of all itervals costitutig {I i }. Give i 2{1,...,}, let S j be as there. The I i = S j2s i J j ad thus I i = j2si J j. Note also that I = S (J j \I) ad {J j \I} I is disjoit. The oly part of that proof that eeds a chace is thus (2.17) which becoes I = J j \ I apple J j apple j2s i J j = I i, (2.19) where the first iequality coes fro the fact that I \ J apple I wheever I,J 2 I the sidelegths of I \J are at ost those of I while the secod iequality follows fro the fact that for each j there is at least oe i such that j 2 S i. Proof of Propositio 2.3. Let us first assue that both {I i } ad {J j } are disjoit collectios with equal uios. The {I i \ J j : i = 1,...,, j = 1,...,} is also a disjoit collectio of half-ope boxes with the sae uio. The by (2.4). Hece I i = I i = I i \ J j ad J j = I i \ J j = I i \ J j = I i \ J j (2.20) J j. (2.21) This proves the clai i the case whe {J j } are disjoit ad the uios are equal. For the case whe {J j } are ot ecessarily disjoit ad the uio of {I i } is oly icluded i the uio of {J j }, we use (2.18) istead of (2.4). 2.2 Outer/ier cotet, easurability. We are ow ready to proclai [ {I i } I disjoit ) c I i := I i. (2.22)

5 14 to be the cotet of ay eleetary set. (The idepedece of represetatio follows fro Propositio 2.3.) A trivial cosequece of the defiitio is E,F 2 E disjoit ) c(e [ F)=c(E)+c(F). (2.23) I order to exted the otio of cotet to ore geeral sets, we ivoke approxiatios: Defiitio 2.6 (Ier/outer cotet, easurability) (1) the outer Peao-Jorda cotet c? (A) of A by c? (A) := if (2) the ier Peao-Jorda cotet c? (A) of A by c? (A) := sup I i : {I i } I AND Let A R. The we defie: [ I i A, (2.24) I i : {I i } I disjoit AND [ I i A. (2.25) (3) We say that A is Peao-Jorda easurable if c? (A)=c? (A). I such a case we call c(a) := c? (A)(=c? (A)) (2.26) the Peao-Jorda cotet of A. LetJ deote the class of Peao-Jorda easurable sets. First we observe that the apparet discrepacy (use of disjoit collectios) betwee the defiitio of outer ad ier cotet plays little role: Lea 2.7 Requirig that the failies {I i } are disjoit i the defiitio of c? (A) leads to the sae value of the ifiu. I particular, for ay A R d we have c? (A) apple c? (A). (2.27) Proof. By Propositio 2.3, represetig the uio S J j as a disjoit uio S I i does ot icrease the value of the su of (pre)cotets. Oce disjoit is required also i the defiitio of c? (A), Propositio 2.3 esures that we get a value o saller tha c? (A). We are the able to coclude: Lea 2.8 (Eleetary sets are easurable) the expressio i (2.22). Every A 2 E is easurable ad c(a) agrees with Proof. Let A be eleetary. Usig A itself i the approxiatios leadig to c? (A) ad c? (A) shows, with the help of the lea before, that c? (A)=c? (A) = the value i (2.22). Hece, the outer cotet of A is thus the least value of cotet see i approxiatios of A by eleetary sets fro outside while the ier cotet of A is the largest value of cotet see i approxiatios of A by eleetary sets fro withi, i.e., ad c? (A)=if c(e): E 2 E, A E (2.28) c? (A)=sup c(e): E 2 E, E A. (2.29)

6 Just as for the cocept of Riea itegrability, beig Peao-Jorda easurable the sigifies that these approxiatios result i the sae value. Let us ow collect soe basic cosequeces of the defiitios of outer/ier cotet. The first oe of these is: Lea 2.9 (Positivity ad ootoicity) For ay A, both c? (A) 0 ad c? (A) 0. I fact, A B ) c? (A) apple c? (B) ad c? (A) apple c? (B). (2.30) Proof. The ootoicity follows iediately. The positivity is the a cosequece of /0 2 I, c? (/0 )=0 ad (2.27). This has a trivial corollary that does ot eve eed a proof: Corollary 2.10 (Null sets are easurable) A 2J ad, i fact, B 2J for all B A. If A is a ull set i the sese that c? (A) =0 the Aother, albeit soewhat less direct, cosequece of the defiitios is: Lea 2.11 (Sub/superadditivity) For ay sets A 1,...,A, c? [ A i apple Siilarly, if A i \ A j = /0 wheever i 6= j, the also c? (A i ) (subadditivity of c? ) (2.31) [ c? i c? (A i ). (superadditivity of c? ) (2.32) A Proof. We start with (2.31). I light of positivity, we ay assue that c? (A i ) < for all i; otherwise there is othig to prove. Let e > 0 ad, for each i = 1,..., fid E i 2 E such that A i E i ad c? (A i )+e/ c(e i ), i = 1,...,. (2.33) Sice S S E i A i, fro Lea 2.2, (2.28) ad (2.23) we thus get c? [ [ A i apple c E i = c(e i ) apple c? (A i )+e/ apple e + c? (A i ). (2.34) But e is arbitrary ad so (2.31) follows. The proof of (2.32) is copletely aalogous. It turs out that A R d is ubouded if ad oly if c? (A) =. (It is istructive to write a proof of this fact.) This shows that, rather tha for subsets of all of R, the Peao-Jorda cotet should be cosidered for subsets of a (easurable) bouded set oly. Here we will fid aother cosequece of the above defiitios quite hady: Lea 2.12 For ay bouded B 2J ad ay A B, I particular, c? (A)=c(B) c? (B r A). (2.35) c? (A) c? (A)=c? (B r A) c? (B r A). (2.36) 15

7 16 Proof. Fix e > 0 ad let E,F 2 E be such that E B (this is where we eed that B is bouded) ad F A obey c(f) c? (A) e. As E r F 2 E by Lea 2.2 ad E r F B r A, fro (2.23) we have c(e)=c(f)+c(e r F) c? (A) e + c? (B r A). (2.37) Varyig E alog a sequece such that c(e) decreases to c? (B)=c(B), we get that apple i (2.35). For the opposite iequality, give e > 0, we pick E,F 2 E such that E B ad F B r A (agai usig the boudedess of B) obeys c(f) apple c? (B r A)+e. The E r F A ad, sice E r F 2 E, fro (2.23) we get c(e)=c(f)+c(e r F) apple c? (B r A)+e + c? (A). (2.38) Takig E alog a sequece such that c(e) teds to c? (B)=c(B), we get coclusio the follows by applyig the first oe to A ad B r A. i (2.35). The secod Note that goig via copleets could be cosidered as a alterative way to defie ier cotet. (We will see that this is exactly what is doe for the Lebesgue easure.) With these properties established, we ow get: Theore 2.13 The followig holds: (1) If A 1,...,A 2J are disjoit, the also S A i 2J ad [ c A i = (2) If A,B are bouded ad easurable, the so is A \ B ad B r A. c(a i ). (2.39) I particular, for ay bouded D 2J, the class {A 2J : A D} of Peao-Jorda easurable subsets of D is a algebra (of subsets of D) ad c is a additive set fuctio o it. Proof. To get (1) we ote that, by Leas 2.7 ad 2.11, [ c? (A i ) apple c? A i apple c? [ A i apple c? (A i ). (2.40) The left ad right-had sides agree wheever A 1,...,A 2J. The clai the follows fro the defiitio of easurability. For (2) let D 2J be bouded ad ote that, by Lea 2.12, A D is easurable if ad oly if D r A is easurable. Now, for A, B D, A \ B = D r (D r A) [ (D r B) A r B = A \ (D r B) thus show that if A,B are easurable, the so are A \ B ad A r B. (2.41) Reark 2.14 We ote that all results i this sectio use oly soe very basic facts about eleetary sets. Naely, we eed that E is closed uder uios, itersectios ad set differeces ad the cotet is well-defied, fiite ad fiitely additive o E.

8 No-easurable sets, relatio to topology. A atural questio to ask is whether there are o-easurable sets i.e., those that fail to be easurable. The aswer to this is quite easy: Lea 2.15 The set A := Q \ [0,1] is ot Peao-Jorda easurable. Proof. The set A := Q \ [0, 1] is totally discoected ad so it cotais o o-trivial itervals. Hece, we get c? (A) =0. O the other had, ay eleetary set that cotais A ust cotai [0,1] ad so c? (A) c? ([0,1]) = 1. A slightly ore subtle questio is what properties of sets characterize Peao-Jorda easurability. A siple exercise shows: Lea 2.16 Let A R d be bouded. The A 2J if ad oly if for each e > 0 there are E,F 2 E with F A E ad c? (E r F) < e. Ufortuately, this is still very uch i the spirit of the origial defiitio. A ore iterestig characterizatio arises whe we try to relate easurability to topology. Recall that, for ay set A R d, A deotes the closure of A while A deotes the iterior of A with respect to the usual Euclidea etric (ad thus topology) o R d. The boudary A of A is the give by A := ArA. Propositio 2.17 (Topological characterizatio of PJ-easuability) For ay bouded A R d, c? (A)=c? (A) ad c? (A)=c? (A ) (2.42) ad so A is easurable if ad oly if c? (A) =c? (A ). Hece, if A is bouded ad easurable the so is ay B such that A B A ad c(b)=c(a) for all of these. I particular, A 2J ) c(a)=c(a)=c(a ). (2.43) Fially, for A R d bouded, Peao-Jorda easurability is characterized by A 2J, c? ( A)=0. (2.44) Proof. We begi by provig (2.42). Let E 2 E be a eleetary sets such that A E ad c? (A)+e c(e). Writig E as the disjoit uio S I i of half-ope boxes {I i }, we ow fid (by elargig I i slightly i all directios) half-ope boxes Ii 0 such that I i I 0 i ad I 0 i apple I i + e/. (2.45) The fact that A is the sallest closed set cotaiig A iplies [ [ A I i Ii 0 = E 0 (2.46) ad we thus get c? (A) apple I 0 i apple I i + e/ = e + c(e) apple c? (A)+2e. (2.47) The ootoicity with respect to iclusio the yields c? (A)=c? (A). The secod part of (2.42) is proved aalogously: We fid a eleetary F A with c? (A) e apple c(f), write it as a disjoit uio of half-ope boxes {I i }, shrik these to half-ope boxes Ii 0 (I i ) with close value of

9 18 cotet, use that A is the largest ope set cotaied i A ad the apply a siilar calculatio as i (2.47) to coclude c? (A ) c? (A). Mootoicity the fiishes the clai. Fro (2.42) we the iediately get that A 2J if ad oly if c? (A)=c? (A ). The (2.43) follows fro the ootoicity of ier/outer cotet with respect to iclusio. Siilarly we get that, oce A 2J, A B A iplies B 2J as well. It thus reais to prove the equivalece (2.44). Lea 2.16 gives the iplicatio ): If E,F are as i the lea, the A E rf ad so c? ( A) apple c(e r F), which ca be ade as sall as desired. For the opposite iplicatio, (, we use c? ( A)=0 to fid a o-epty F 2 E such that A F ad c(f) < e. The A r F A. But A r F is closed ad so oe ca fid E 2 E such that But E [ F 2 E ad E [ F A. Therefore, A r F E A ad c? (A) e c(e). (2.48) c? (A) apple c(e [ F) apple c(e)+c(f) < c? (A )+2e. (2.49) Hece, c? (A)=c? (A ) ad, by the first part of the clai, A 2J. 2.4 Coectio with Riea itegral. The fact that easurability of A iplies easurability of both A ad A sees quite atural: ideed, we usually expect that the closure ad iterior of a set are ore regular tha the set itself. However, this coclusio has its liitatios: Lea 2.18 There exists a closed set A [0,1] which is ot Peao-Jorda easurable. Proof (sketch). Oe exaple for this would be ay fat Cator set i [0,1]. (The existece of such sets is what uderlies Lea 1.8.) Eve ore siple is a exaple costructed as follows: Let {q } euerate Q ad, give e > 0, let A :=[0,1] r [ 1(q e2,q + e2 ). (2.50) It is obvious that A is closed (possibly epty). By trucatig the uio at a fiite ad replacig ope itervals by half-ope oes, we readily fid c? (A) 1 2e. O the other had, A is totally discoected ad so c? (A)=0. So for e < 1 / 2, A is a o-epty closed set which is ot Peao- Jorda easurable. The coclusio that the cotets of A, A ad A are all equal was, at the tie of coceptio of this theory, cosidered as copletely atural. However, later it was foud too restrictive ad, i fact, a liitatio of the theory. Here we will deostrate this further by statig without proof two leas that show that easurability is essetially equivalet to Riea itegrability. Lea 2.19 Let A R d be a bouded set. The A 2J if ad oly if 1 A, the characteristic fuctio of A, is Riea itegrable. Lea 2.20 Let a,b 2 R obey a < b ad let f : [a,b]! [0, ) be a fuctio. The f is Riea itegrable, (x,t) 2 [a,b] [0, ): f (x) apple t 2J (2.51)

10 Uderlyig these is the fact that the Peao-Jorda cotet is ot coutably additive. obstructio is explaied i the ext lea, whose proof we also leave to the reader: Lea 2.21 There are sets A 1,A 2, 2J with A i 2 [0,1] for all i such that S A i 62J. (Note, however, that if the uio is easurable, ad the sets A i are disjoit, the cotet is coutably additive o this sequece.) The puchlie is that, despite all of the coceptual developets, we still have ot left the paradig (ad thus have to face also the associated shortcoigs) of Riea itegrability. However, as we will see i the ext chapter, we are already o a good way to do so. 19 The