实变函数外测度

1、LI MIAO: LECTURES ON REAL ANALYSISChap ter 3. Measure TheoryThe Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volumes to subsets of Euclidean space It is used throughout real analysis, in particular to dofiiie Lebesgue integration Sets which can be

2、assigned a volume are called Lebesgue measurable; the volume or measure of the Lebesgue measurable set A is denoted by m(A). A Lebesgue measure of oc is possible, but even so. assuming the axiom of choice, not all subsets of are Lebesgue measurable.Henri Lebesgue described his measure in 1901, follo

3、wed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.Examples.1. If E is a closed interval a, 6, then its Lebesgue measure is the length b a The open interval (a, 6) has the same measure, since the difference between the two sets has

4、measure zero.2. If E is the Cartesian product of intervals a. b and c. d, then it is a rectangle and its Lebesgue measure is the area (b a)(d c).3 The Cantor set is an example of an uncountable set that has Lebesgue measure zeroThe modern construction of the Lebesgue measure, based on outer measures

5、, is due to Caratheodory.LI MIAO： LECTUKES ON REAL ANALYSIS#§1. Volumes of open setsFix n e N. An open cube in Rz? is a set of the form I =%), where bi > % We callme-othe volume of I.Lemma 1 If G is an open set in then G is the union of countable disjoint cubes of tlie form (©,%.Lemma 2

6、 Letand be two families ofcubes. Ifare disjoint and U匙厶 CthenLemma 3. If G is an open set in IR" and G = UA- = UJ/s where 1也 Jk are disjoint cubes of the formthenHk 11 =1*1-It is therefore reasonable to define the volume of G byooG = 52 囚k=lif G = UaJa-. where Ik are disjoint cubes.Properties.

7、Let G and Gk be open sets in 展".(1) If G0, then |G| > 0:(2) If Gi C G2, then |Gi| < |G2|;(3) | U直1 Gk < EXi I6|；If Gk are disjoint then I U畳】Gk =工Xi |6|.LI MIAO： LECTUKES ON REAL ANALYSIS5§2. Out er measuresOuter measures. Let £7 be a set in Rn. we define the outer measure of

8、 E bym*(E) = inf|G| : G D E and G is open.It is equivalent to sayoooon=ln=lIk is an open cube for every kwhereIk = (1? 卫 J : <2/ < t= 1, , nwithn|4| = ("-©)i=lExamples.1. If £?=«, then m*(F) = 0.2. If E is at most countable then m*(E) = 0.3. If / is a cube, then m*(7) = |/|

9、.We first show that if / is a closed cubebj, thenm*(/) =IK=-a)For every e > 0, the open cube U=i(a» 一 6 ® + e) coversI、thusnm*(/) </ - ai + 2e),1=1letting >() we havenm* <- ai).i=lNext we show the converse inequality. To this end we show that for every finite or countable open

10、covering of 厶 占砂,ny? i-Gi 2 ip® -如.jeJi=iSince / is a bounded closed subset of by the Heine- Borel theorem, there is a finite subcovering of Ij such that Ujlj D I By Lemma 2 in §1, the above inequality holds.4. i*(G) = 0, where G is the Cantor set.Properties 1. (Positivity) m*(E) > 0, m

11、*(0) = 0.2. (Monotonicity) If A C B, then m*(X) <3. (Countably subadditivity) m*(|J二i EJ < 刀二i4. (Countably additivity under separation condition ) If there are disjoint open sets Gh such that Ek C then m*(U亀 Ek) = EZi m*(EQ.5. (Translation-invariant) m*(£? + t) = m*(£?).However, the

12、 outer measure 771* is not countably additive, that is, if An are disjoint, the identitynis not true in general.An example (of a set which is not measurable)Define an equivalent relation on (0,1):s y <=> = y CThen the interval (0.1) can be decomposed into some dis- joiiit equivalent classes By

13、 axiom of choice, from each equivalent class of (0,1)/ we can take an element Let S be the set of all these elements. It is obvious that S C (0.1).Let c豊i = (-1,1) AQ and Sn = rn + S := rn + x : x S. Here are some properties of Sn:1. Sn C (1,2) for every n.2. = m*(5).3. Sn D Sm = 0 if 72 # m.4. (O.l

14、)cUi.For such family of disjoint sets S,LI MIAO： LECTUKES ON REAL ANALYSIS1§3 Measurable sets and measuresMeasurable sets. A set E C 1R" is called (Lebesgue) measurable if for any set T U 展"，m*(T) = m*(T Q E) + m*(T n Ec)and 77?(E) := z?T(E) is called the measure of E.Properties 1. E

15、is measurable if and only if Ec is measurable.2. If m*(£,) = 0, then E is measurable.3. 0 and IR" are measurable.4. If Ei, £*2 are measurable, then E1UE2, E1QE2, E E? arc measurable.I. We first show that if E is measurable and A U E、B U Ec, thenmA US) = mA) + m*(5).II. Now for every s

16、ubset T、m*(T) < mT A (Ei U E2) + m(T n (£；i U E2)c)= m*(r n Ei)u (r n e2) + m(T n n= m*(r n e2)n Ei u |(r n %)n ex u(rn e2)+m*(rn£；f n)=m*(TQE$)QEiU0QE2)+m*(rnEf nE$)= m*(T n E n El) + m*(T n f2)= m*(rn)+ m*(rn£；2)=m*(r)5. If Ei is measurable for each i, then U秽 1 E)is measurable .

17、If in addition. Ei are disjoint, thenococm(Ud)=工 m(EJ.i=li=lWithout loss of generality, we can assume that Ei are disjoint. By induction on n, we havennm(U Ei) = Zm(EJ.i=li=lSince U： 耳 is nieasurable, for eveiy subset T,nnm*(T) = mT A (|J E,) + m*(T A (|J Ef)c) 1=11=1noo> m*(T A (J E,) + mT n (|J

18、 E,)c)i=li=lnoo=m*(T A Kf) + mT n (|J Et)c)1=1t=lLet ting n > oc we have by the countably subadditivity of outer measures,oooom*(T) < m*(r n (|J Ei) + m*(T n (|J Et)c)1=1J=1oooo< 刀加 ctqe)+ 7心fci(UEy)i=li=l< m*(T)6. If Ei is a family of increasing measurable sets, then limioc Ei is measur

19、able andm( lim Ej) = lini m(E).i>oci>ooSuppose that liin/oc m(Ei) < +00、then00= m(EiuU(E+i-E)ii=l00m(E*i) + ym(Ej-n) 一 m(Ez) = liin m(EJ.1=17. If 耳 is a family of decreasing measurable sets, then limioc E is measurable. And if there is some io such 771 (EJ < +00, thenm( lim Ei) = lim m(E

20、).i>oc1>008 If Ei is measurable for each i and limoc Ej exists, then it is measurable. And if there is some K()such that m(Uy° EJ < +00, thenm( lim Ej) = lim m(尽).«>oc1*ooNow let £(IR") be the set of all Lebesgue measurable sets on Rz,. Then1. £(RZ,) is a(7-ficld

21、.2. = 2C.3. If m*(E) = 0, then E £(Kn).Construction of Lebesgue measurable setsConstruction:1. If I is an open cube of R/l, then I is measurable and m(/) = |/|.Let I =®).a. The half space(b ,rrn) : rrn > 0LI MIAO： LECTUKES ON REAL ANALYSIS11is measurableb. The set( 1，7i) : Qn V V 九is me

22、asurable.c. The setEi := (工1,岛)：ai < Xi < biis measurable for every z = L n.7JcL / = Ei is measurable.2. If G is an open set in then G is the union of at most countable open cubes.An open cube I =is called a rational cubeif aj. bj E Q for all i. We claim that for every open ball r), there is a

23、 rational cube I C r) and x G I. Indeed, let x =(? xn For 1 < z < n, we can find ai, bi e Q such thatrrXi< ai < xi < bi < xi-nnIt is obvious that I = 11二1(%®) is a rational cube containing x. Moreover, I C O(x. r) (exercise)3. All open sets，closed set, Fa sets and sets arc mea

24、surable.A countable intersection of open sets is called a G& set; a countable union of closed sets is called an Fa set4 Borel measurable sets: the smallest(7-field of containing all open sets.Let Z?(Rn) be the set of all Borel measurable sets on Rz Then(a) jB(Rn) is a a-field.(b) 3(时)C £(Rn

25、).(c) 3(曲)=C(d) All open sets, closed sets, Fa sets and G& sets are Borel measurable5. If E is a Lebesgue measurable set with m(E) = 0 (a null set), then every subset of E is also a null set. A fortoiori, every subset of E is measurable.If E is a Lebesgue measurable set, then it is uapproxi- lii

26、ately closed and approximately openr in the sense of Lebesgue measure:Theorem. Let S be a Lebesgue measurable set on Then for every 6 > (), there exist subsets F and G of IRn such thnt1. F is closed;2. G is open;3. F U E U G;4. mG - F) < e.Proof. I. First we assume that E is bounded. For every

27、 e > 0, we can find an open set G such that E U G and772(G) < m(E) + e/2.Thusm(G E) = m(G) -< e/2.II. Suppose that E is unbounded. There is a sequence of cube Jh such that Jk C Jk+i and |JA. Jk = IR7i. Let Ek = E C 几 then Ef. is bounded Therefore, for every 札 we can find an open set such th

28、at C O斤 and 771(0 Ek) < e/2A+1. Let O = |J 0比 then O is open,E = jEkcJOk = Okkandm(O - E) = m(|J O,. - E) < m( J(O, - E,)kk<- EJ V “2.kNow let U be a open set such that m(U Ec) < e/2. Let F = Uc, then F is closed such that F U E andm(E F) = mE Uc) = m(E A t/) = m(U - Ec) < e/2.Theorem

29、. Let E be a Lebesgue measurable set on then(a) there is a set G Z) E such that m(G E) = 0.there is an Fa set F (Z E such that m(E F) = 0.LI MIAO： LECTUKES ON REAL ANALYSIS17§4. Product spacesTo begin with, we define a rectangle to be set of the formA x B = (t, y) : x E A. y E B,where A C Rn an

30、d B CLet Cn. Cm, Cnm be the set of Lebesgue measurable sets on IRn,IRm?IRn+m, respectively.Theorem. If X G Cn and B G Cm, then A x B Moreover,mA x B) = m(A)m(73),with the understanding tliat if one of the sets A and B has measure zero, then m(A x B) = 0.Proof. The proof is divided into several steps

31、.I. B are cubes.In this case A x 13 is also a cube and m(A x B)= m(A)77i(Z?).II. A. B are open sets.There are disjoint cubes 4 Bj such that X = |J£B = U宀A x B = |J(A x BjY andmA x 13) =m(4 x Bj) = m(AJm(I3j)ij=(工 (工 m(巧)=m(A)m(B).III. A, B are bounded G-sets.One can find decreasing sequences of

32、 open sets G» G7 such that A =B = p|j Gj, thereforeAx B = p|(G, x Gi)iis measurable andmA x B) = lim m(Gi x Gi) =i*ooIV. A. B are bounded measurable sets.There are bounded G$-sets G and G such thatG D A with mG X) = 0, G D Z? with m(G Z?) = 0.Note thatG x G = (A x /?) U (G - A) x G U G x (G - $

33、), we only need to show that both (G-A)xG and Gx (GB) has measure zero.V. A. B are measurable sets.Let Ak = AD x E : x < k and Bk = BDy E R7Z,: y < k, then Ak C AM,凤 U and4 = U 几 =U %kkSince Ak. Bk are bounded measurable sets, Ak x Bk are measurable and thus A x B = Jk x is measurable. Moreove

34、r,m(A x B) = limx B)A:>oo=lim m(Ak)m(I3k) = m(4)m(J3) k>ooIf at least one of A and B has measure zero, then all the sets Ak x Bk has measure zero, so that m(A x Z?) = 0.It is t herefore reasonable to call Ax 13 measurable rectangle if both A and B are measurable Clearly(4 x B) n (E x F) = (>

35、;1 n F) x (B n F)(A x B)c = (Rn x Bc) U (Ac x B).Therefore, the collection of finite disjoint union of measurable rectangles is a field, and the cr-field it generates is denoted by Cn x Cm.Relations between £n+/n and Cn x1.2. 3(时+")c cn x £m.3 For every E there exist E. E? G J x £

36、;msuch that E U E U E? and m(E、2 E) = m(E EJ = 0.If E U Rz,+m, for t X and y E Y we define the x- section Ex and the y-section Ey of E byEx = ye :(,“) E, Ey = x e : (x, y) e E.Theorem. If E E Cn x then for every x y e 邮 Ex e E" e Cn.Proof. Let be the collection of all subsets E of Cn x Cm such

37、that Ex G Cm for all x and Ey G Cn for all y. Then g obviously contains all measurable rectangles. Since (|J Ej)x = |J(Ej)z and (Ec)x = (EJ, and likewise for “-sections, £ is a cr-field Therefore, D £n x Cin.Theorem. If F £n+rn with 772(E) = 0. then for almost all x e RZA,= 0.Theorem.

38、 If E Ethen for almost every x G§5. Abstract measuresA measure space consists of a set X equipped with two fundamental objects:(1) A rr-field M of Umeasurable sets, which is a nonempty collection of subsets of X closed under complements and countable unions and intersections.(2) A measure /z :

39、jM 0. +oo with the following defining property: if E?、 is a countable family of disjoint sets in 丿" thenococ“(U En) =卩(En).n=ln=lA measure space is therefore often denoted by the triple to emphasize its three main components.X is called(7-finite if it can be written as the union of countably ma

40、ny measurable sets of finite measure.Examples 1. X = %醫i is a countable set, M is the collection of all subsets of X，and the measure “ determined by “(©) = with "豐i a given sequence of (extended) lion-negative numbers. Note that “(E)= 疋eWhen“n = 1 for all 72, we call “ the counting measure

41、 Ill this case integration will amount to nothing but the suinination of convergent series.2 Let X = M, be the collection of Lebesgue measurable sets, and “(E) = JE fdx where / is a given nonnegative measurable function on IRn. The case f = 1 corresponds to the Lebesgue measure.Outer measures and Ca

42、ratheodory theorem.Definition. Let X be a set. An outer measure on X is a function from the collecticm of all subsets of X to 0. oc that satisfies the following properties:心)=()(ii) If Ei C 场,then 讥EJ < /*(E2).(iii) If E, £?2, is a countable family of sets, thenoooc”(U顷)辽以跌)k=lk=lA set E in

43、X is Caratheodory measurable or simple measurable if one has“*(4) = Q E) + "(A A E) V4 C X.Theorem. Given an outer measure /* on a set X, the collection M of Caratheodory measurable sets forms a a- field. Moreover,restricted to A4 is a measureAs we have seen, a class of measurable sets on X can be construe tod once we start, with a given outer measure. However, the definition of an outer measure usually depends on a more primitive idea of measure defined on a simpler class of sets. This is the role of a premeasure defined belowLet X be set. A field in X is a non-einp