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Unformatted text preview: Chapter 0
Measures, Integration, Convergence 1. Measures
Deﬁnitions
Basic Examples
Extension Theorem
Completion 2. Measurable Functions and Integration
Simple functions
Monotone convergence theorem (MCT)
Fatou’s lemma
Dominated convergence theorem (DCT)
Absolute continuity of the integral
Induced measures
Theorem of the unconscious statistician 3. Absolute Continuity, RadonNikodym Theorem, Fubini’s Theorem
Absolute continuity Radon—Nikodym Theorem
Fubini—Tonelli Theorem Chapter 0 Measures, Integration, Convergence 1 Measures Let Q be a ﬁxed non—void set. Deﬁnition 1.1 ( ﬁelds, U—ﬁelds, monotone classes) A non—void class A of subsets of Q is
called a: (i) ﬁeld or algebra if A, B E A implies A U B E A and AC 6 A.
(ii) U—ﬁeld or U—algebra if A,A1,A2, . . . E A implies UfoAn E A and AC 6 A.
(iii) monotone class if An is a monotone / (\) sequence in A implies UfoAn E A (FROAH E A). (iv) (Q,A) With A a U—ﬁeld of subsets of Q is called a measurable space. Remark 1.1 A,B E A imply A H B E A for a ﬁeld. (ii) A1, . . .,A,,, . . . E A implies ﬂﬁleAn E A for a U—ﬁeld. (iii) (1), Q E A for both a ﬁeld and U—ﬁeld. (iv) To prove that A is a ﬁeld (U—ﬁeld) it sufﬁces to show that A is closed under complements and
ﬁnite (countable) intersections. Proposition 1.1 Arbitrary intersections of ﬁelds (U—ﬁelds) ((monotone classes)) are ﬁelds
(U—ﬁelds) ((monotone classes)).
(ii) There exists a minimal ﬁeld (0— ﬁeld) ((monotone class)) 0(C) generated by any class of subsets of 9. (iii) a U—ﬁeld is a monotone class and conversely if it is a ﬁeld. Proof. (iii) (<2) ugozlAn : Ugo:1(UZ:1Ak) E Ufan where B,, /. Notation 1.1 If 9 is a set, 29 is the family of all subsets of Q.
20 is always a U—ﬁeld. Example 1.1 If 9 : R, let BO consist of(1) together With all ﬁnite unions of disjoint intervals of
the form U?:1(a,,1),], or U?:1(a,,1),] U (a,,+1,oo), (—oo, 1),,+1] U U?:1(a,,1),], With (1,, 1), E R. Then 80
is a ﬁeld. 4 CHAPTER 0. MEASURES, INTEGRATION, CONVERGENCE Example 1.2 If 9 : (0,1], let BO consist on) together with all ﬁnite unions of disjoint intervals
of the form U?:1(a,,b,], g a, < b, g 1. Then 80 is a ﬁeld. But note that 80 does not contain intervals of the form [(1,1) or ((1,1)); however ((1,1)) : og°=1(a,b — Example 1.3 If 9 : R, let C : 80 of example 1.1, and let 8 be the U—ﬁeld generated by 80;
B : 0(80). 8 is a U—ﬁeld which contains all intervals, open, closed or half—open. From real analysis,
any open set 0 C R can be written as a countable union of (disjoint) open intervals: 0 : Ugo:1(an, bn). Thus 8 contains all open sets in R. This particular 8 E 81 is called the family of Borel sets. In
fact, 8 : 0(0), where O is the collection of all open sets in R. Example 1.4 Suppose that Q is a metric space with metric ,0. Let (9 be the collection of open
subsets of Q. The the U—ﬁeld B : 0(0) is called the Borel U—ﬁeld. In particular, for Q : Rk with the Euclidean metric p($,y) : $ — y : — yi2}1/2, B E Bk E 0(0) is the U—ﬁeld of Borel
sets. Deﬁnition 1.2 A measure (ﬁnitely additive measure) is a function ,u : A —> [0, 00] such that
,u((Z)) : 0 and ME An) : ZMAn) for countable (ﬁnite) disjoint sequences An in A.
(ii) A measure space is a triple (Q,A,,u) with A a U—ﬁeld and ,u a measure. Deﬁnition 1.3 ,u is a ﬁnite measure if 11(9) < 00. (ii) ,u is a probability measure if ,u(Q) : 1. (iii) ,u is an inﬁnite measure if ,u(Q) : 00. (iv) A measure ,u on a ﬁeld (U—ﬁeld) A is called U—ﬁnite if there exists a partition {Fn}n21 C A such that Q : F7, and ,u(Fn) < 00 for all n 2 1.
(v) A probability space is a measure space (Q,A,,u) with ,u a probability measure. Deﬁnition 1.4 A measure ,u on (Q, A) is discrete if there are ﬁnitely or countably many points
to, in Q and masses m, E [0, 00) such that ,u(A) : E m, for A E A.
wiEA (ii) If ,u is deﬁned on (9,29), 9 arbitrary, by ,u(A) : ﬁof points in A, ,u(A) : 00 if A is not ﬁnite,
then ,u is called counting measure. Example 1.5 A discrete measure ,u on (Q,A) : (R1,81): x, : i, m, : 2i. (ii) A discrete measure ,u on (Q,A) : (Z+,QZ+): x, : 22,771, : 1/2'. (Z+ : {1,2, . . (iii) Counting measure on (R1,81); not a U—ﬁnite measure! (iv) Counting measure on (Z+, 22+). (v) A probability measure on Q, the rationals: With an enumeration of the rationals, let
m, : 6/(7r2i2). Proposition 1.2 Let (Q,A,,u) be a measure space.
(i) If {An}n21 C A with An C An+1 for all n, then ,u(U%°=1An) : limn_,OO ,u(An).
(ii) If ,u(A1) < 00 and An 3 An+1 for all n, then ,u(ﬂ,‘3,°:1An) : limn_,OO ,u(An). 1. MEASURES 5 Proof. M(U?°An) M(U?°(An \ 1471—1)) Where A0 :0 E 2 MAR \ An_1) by countable additivity
1 TL = ligl Zu<An\An_1) lim M(Bn) = M A1) — MﬂfoAn) by ﬁnite additivity,
while on the other hand,
li7m ,u(Bn) : liqrbn ,u(A1 \ An) : li7m{M(A1) — M(An)} by ﬁnite additivity
= M(A1) — 11,? MAU Combining these two equalities yield the conclusion of (ii). Deﬁnition 1.5 (i) li_mAn E U221 0:02,, Ag, E {w E Q : w 6 all but a ﬁnite number ofAzs} E [An mm];
(ii) EAR E 0,2021 Uzin Ag, E {w E 9 :w E inﬁnitely manyAzs} E [An i.0.]. Remark 1.2 li_mAn C mAn; limAn E li_mAn provided li_mAn : EA”
Proposition 1.3 Monotone / (\) An’s have lim An : UfoAn (2 {KOAR Example 1.6 Let A : B : 0(80) as in example 1.3. For B E 80, let ,u(B) E the sum of the
lengths of intervals A E 80 composing B. Then ,u is a countably additive measure on 80. Can ,u
be extended to B? The answers is yes, and depends on the following: Theorem 1.1 (Caratheodory Extension Theorem) A measure ,u on a ﬁeld C can be extended
to a measure on the minimal U—ﬁeld U—ﬁeld 0(C) over C. If ,u is U—ﬁnite on C, then the extension
is unique and is also U—ﬁnite. Proof. See Billingsley (1986), pages 29 — 35 and 137 — 139. 6 CHAPTER 0. MEASURES, INTEGRATION, CONVERGENCE Example 1.7 (example 1.3, continued.) The extension of the countably additive measure ,u
on 80 to 81 : 0(80), the Boreal U—ﬁeld, is called Lebesgue measure; thus (R1,81,,u) where ,u is
the extension of the Caratheodory extension theorem, is a measure space. The usual procedure is
to complete 81 as follows. Deﬁnition 1.6 If (Q,A,,u) is a measure space such that B C A with A E A and ,u(A) : 0 implies
B E A, then (Q,A,,u) is a complete measure space. If ,u(A) : 0, then A is called a null set. (Of
course there can be non—empty null sets.) Exercise 1.1 Let (Q,A,,u) be a measure space. Deﬁne
XE {AUN:AEA,N C Bfor someB EAsuch that ,u(B) :0} and let ﬁ(A U N) E ,u(A). Then (9,],ﬁ) is a complete measure space. Example 1.8 (example 1.3, continued.) Completing (R1,81,,u) where ,u :Lebesgue measure
yields the complete measure space (R1,El,ﬁ). 8—1 is called the U—ﬁeld of Lebesgue sets. So far we know only a few measures. But we will now construct a whole batch of them; and
they are just the ones most useful for probability theory. Deﬁnition 1.7 A measure ,u on R assigning ﬁnite values to ﬁnite intervals is called a Lebesgue —
Stieltjes measure. Deﬁnition 1.8 A function F on R which is ﬁnite, increasing, and right continuous is called a
generalized distribution function (generalized df). Fm, b] E m) — Fm) for —00 < a < b < 00 is called the increment function of the generalized df F. We identify generalized df’s having the same increment function. Theorem 1.2 (Correspondence theorem.) The relation
,u((a,b]):F(a,b] for —oo<a§b<oo establishes a one—to—one correspondence between Lebesgue—Stieltjes measures ,u on B : 81 and
equivalence classes of generalized df’s. Proof. See Billingsley (1986), pages 147, 149 — 151. Deﬁnition 1.9 (Probability measures on R.) If ,u(Q) : 1, then ,u is called a probability dis—
tribution or probability measure and is denoted by P. Deﬁnition 1.10 An /, right—continuous function F on R such that F(—oo) : 0 and F(oo) : 1
is a distribution function (df). Corollary 1 The relation
P((a,b]):F(b)—F(a) for —oo<a§b<oo establishes a one—to—one correspondence between probability measures on R and df’s. 2. MEASURABLE FUNCTIONS AND INTEGRATION 7 2 Measurable Functions and Integration Let (SLA) be a measurable space.
Let X denote a function7 X : 9 —> R. Deﬁnition 2.1 X : 9 —> R is measurable if [X E B] E X_1(B) : {w E Q : X(w) E B} E A for all B681. Deﬁnition 2.2 For A E A the indicator function of A is the function 1A(w) :{ iwaA
ifuJEAC (ii) A simple function is X(w) E 1 3621141. (w) for Air : 9, AZ' 6 A, 962' E R. 2: (iii) An elementary function is X(w) E $21141. (w) for AZ' : 9, AZ' 6 A, 96¢ E R. Proposition 2.1 X is measurable if and only if X_1(C) E {X_1(C) : C E C} C A Where 0(C) : 8.
Hence X is measurable if and only if X_1((907 00)) E [X > 90] E A for all 90 E R. Proof. (=>) This direction is trivial. (<2) X_1(B) : X_1(0'(C)) : 0(X_1(C)) since X‘1 preserves all set operations and since X_1(C) C
A With A a U—ﬁeld by hypothesis.
Further7 U({(x, 00) : 90 E R}) : 81 since (a,b] : ((1.00) H (b,oo)c7 and 81 is generated by intervals of the form (a, b]. Note that the assertion of the propostion would work With (90, oo) replaced by any Of [$7OO)7 (—007xl7 (—007x)‘ Proposition 2.2 Suppose that {Xn} are measurable. Then so are supn Xn7 —Xn7 infn Xn7 EXR7 li_an7 and lim Xn. Proof. [811an > $l : UanTL > xii [—Xn > x] : [Xn < —x];
inf Xn : — supn(—Xn);
EXR : infn(supk2n Xk); 11—an = —E(—Xn);
limn Xn : EXR When lim Xn exists. Proposition 2.3 X is measurable if and only if it is the limit of a sequence of simple functions: Xn = _n1[X<—n]  2 Proof. 2n 1[(k—1)/2"SX<k/2“] + ”1[X>n1
k:—n2"+1 (=>) The Xn’s exhibited above have Xn(w) — X(w) < 2‘” for X(w) < n.
(<2) The exhibited Xn’s are simple7 converge to X7 and lim Xn is measurable by prop 2.2. Remark 2.1 If X Z 07 then 0 g Xn / X. 8 CHAPTER 0. MEASURES, INTEGRATION, CONVERGENCE Proposition 2.4 Let X,Y be measurable. Then X :: Y, XY, X/Y, X+ E X1[XZO], X‘ E
—X1[XSO], X, g(X) for measurable g are all measurable. Proof. Let Xn, Yn be simple functions, X7, —> X, Yn —> Y. Then Xn :: Yn, XnYn, Xn/Yn
are simple functions converging to X :: Y, XY, and X/Y, and hence the limits are measurable by
prop 2.3. X‘l' and X‘ are easy by prop 2.3, and X : X‘l' —— X‘. For g : R —> R measurable we
have, for B E 81, (gX)_1(B) : _1(g_1(B)) : X_1( a Borel set ) since 9 is measurable X
E A since X‘1 is measurable. Remark 2.2 Any continuous function g is measurable since
g_1(8) : g_1(0((’))) : (ﬁg—1(0)): 0( a subcollection of open sets ) C 8. Now let (Q,A,,u) be a measure space, and leet X,Y denote measurable functions from (Q,A)
to (R,E), R E R U {:00}, E E 0(8 U {00} U {—oo}).
CONVENTIONS:0OO:0:OO0,$00:oo$:ooif0<ac<00;0000200. Deﬁnition 2.3 For X E 36,1,4, with x, Z 0, A, : Q, then fXdM : x,M(A,). (ii) For X Z 0, fXdM E limanndu where {Xn} is any 2 0, / sequence of simple functions,
Xn —> X. (iii) For general X, fXdM E fX‘l'dM — fX‘dM if one of fX‘l'dM, fX‘dM is ﬁnite. (iv) If fXdM is ﬁnite, then X is integrable.
JUSTIFICATION: See Loeve pages 120 — 123 or Billingsley (1986), page 176. Proposition 2.5 (Elementary properties.) Suppose that fXdM, deu, and fXdM —— deM
exist. Then: (i) I(X+Y)du = fXdH‘l'deMa chdM = CfXdu; (ii) X Z 0 implies fXdM Z 0; X Z Y implies fXdM Z deM; and X : Y a.e. implies fXdM :
deM. (iii) (integrability). X is integrable if and only if X is integrable, and either implies that X is a.e.
ﬁnite. X g Y with Y integrable implies X integrable; X and Y integrable implies that X —— Y is
integrable. Proof. (iii) That X is integrable if and only if fX‘l’dM and fX‘dM ﬁnite if and only if
X integrable is easy. Now fX‘l'du < 00 implies X+ ﬁnite a.e.; if not, then ,u(A) > 0 where
A E {w : X+(w) : 00}, and then fX‘l'dM Z fX+1Ad,u : oo ,u(A) : 00, a contradiction. Now
0 g X+ g Y, thus 0 g fX‘l’dM g deM < 00. Likewise fX‘du < oo. Theorem 2.1 (Monotone convergence theorem.) If 0 3 X7, / X, then andlu —> fXdM. Corollary 1 If X7, 2 0 then [220:1 Xndu : 220:1 andM. 2. MEASURABLE FUNCTIONS AND INTEGRATION 9 Proof. Note that 0 g Xk / Xk and apply the monotone convergence theorem. Theorem 2.2 (Fatou’s lemma.) If X7, 2 0 for all n, then fli_and,u g Xndu . Proof. Since X7, Z inka,, Xk E Y7, / li_an, it follows from the MCT that /li_and,u : /limYnd,u : lim/Yndu g li_m/Xnd,u. Deﬁnition 2.4 A sequence X7, converges almost everywhere (or converges a.e. for short), denoted
X7, —>a.e. X, if X,,(w) —> X(w) for all to E 9 \ N where MN) : 0 (i.e. for a.e. to). Note that
{Xn}, X, are all deﬁned on one measure space (9,A). If ,u is a probability measure, ,u : P with
P(9) : 1, we will write —>a.5. for —>a.e.. Proposition 2.6 Let {Xn}, X be ﬁnite measurable functions. Then [Xn —> X] : 020:1 U221
ﬂgfanXm — XI < 1/k], and is a measurable set. Corollary 1 Let {Xn}, X be ﬁnite measurable functions. Then X7, —>a.e. X if and only if
limit; U232. lle — XI 2 6) = 0
for all 6 > 0. If M9) < 00, X7, —>a.e. X if and only if
M(U:,O:n[Xm — X Z 6) —> 0 as n —> 00 for all 6 > 0. Proof. First note that an —> ch : U20:1 ﬂ720:1 Ufnoznlle — XI 2 1/kl E UzO=1Ak with Ak /; and Ag, : D” Em with Bnk \ in n. Applying prop 1.2 gives the result. 7121 Deﬁnition 2.5 (Convergence in measure; convergence in probability.) A sequence of ﬁ—
nite measurable functions X7, converge in measure to a measurable function X, denoted X,, —>,) X,
if Mlan — XI 2 6]) —> 0
for all 6 > 0. If ,u is a probability measure, M9) : 1, call ,u : P, write X7, —>,, X, and say X7,
converge in probability to X. Proposition 2.7 Let Xn’s be ﬁnite a.e.
(i) If X,, —>,, X then there exist a subsequence such that Xnk —>a.e. X.
(ii) If M9) < 00 and X7, —>a.e. X, then X,, —>,, X. Theorem 2.3 (Dominated Convergence Theorem) If Xn g Y a.e. with Y integrable, and
if X,, —>,, X (or X,, —>a.e. X), then f Xn — Xdlu —> 0 and limendM : fXdM. 10 CHAPTER 0. MEASURES, INTEGRATION, CONVERGENCE Proof. We give the proof under the assumption Xn —>a.e. X. Then Zn E Xn — X —> 0 a.e.
and Zn 3 Xn —— X g 2Y E Z. Thus Z — Zn 2 0 and by Fatou’s lemma /Zd,u : — Zn)d,u g li_m (Z — Zn)d,u : /Zd,u —E/an,u,
and this implies
E/anﬁ/pfn—qugo. Thus
/Xt—/XW=t/ua—Xymbs/mg—quso Deﬁnition 2.6 Let X be a ﬁnite meaurable function on a probability space (Q,A,P) (so that
13(9) : 1). Then X is called a random variable and .ﬁﬂDEHXEBﬁ:HwEQpﬂMEBD for all B E B is called the (induced) probability distribution of X (on R). The df associated With
PX is denoted by FX and is called the df of the random variable X. Thus (R, B, PX) is a probability
space. Theorem 2.4 (Theorem of the unconscious statistician.) lfg is aﬁnite measurable function
from R to R, then hémenwmn=/ R mmﬂﬂm:émmﬂﬂm Proposition 2.8 (Interchange of integral and limit or derivative.) Suppose that X(w,t) is
measurable for each If 6 ((1,1)). (i) le(w,t) is a.e. continuous in t at to and X(w,t) g Y(w) a.e. for t—tol < 6 With Y integrable,
then fX(,t)d,u is continuous in t at 150. (ii) Suppose that %X(w,t) exists for a.e. to, all t 6 ((1,1)), and %X(w,t) g Y(w) integrable a.e.
for all t 6 ((1,1)). Then géXmmmw=(%Mwwma Proof. (ii). By the mean value theorem
X(w,t——h) — X(t) _ 8
———7————EMWW3 for some t g .9 3 15+ h. Also the left side of the display converges to %X(w,t) as h —> 0 for a.e. w,
and by the equality of the display and the hypothesized bound, the difference quotient on the left
side of the display is bounded in absolute value by Y. Therefore g/meww):1ml{/MWHWMMM—/XWMWWﬁ h—>0h
_ ' X(w7t+h)_X(w7t)
— {fw : /%meww) Where the last equality holds by the dominated convergence theorem. 3. ABSOLUTE CONTINUITY, RADON—NIKODYM THEOREM, FUBINI’S THEOREM 11 3 Absolute Continuity, RadonNikodym Theorem, Fubini’s The
orem Let (Q,A,,u) be a measure space, and let X be a non—negative measurable function on 9. For A69, set
1/(A)E/Xd,u:/1AXd,u.
A 9 Then 1/ is another measure on (Q,A) and 1/ is ﬁnite if and only if X is integrable (X E L1(,u)). Deﬁnition 3.1 The measure 1/ deﬁned by 7? is said to have density X With respect to 11. Note that ,u(A) : 0 implies that 1/(A) : 0. Deﬁnition 3.2 If ,u, 1/ are any two measures on (Q,A) such that ,u(A) : 0 implies 1/(A) : 0 for
any A E A, then 1/ is said to be absolutely continuous With respect to ,u, and we write 1/ << ,u. We
also say that 1/ is dominated by 11. Theorem 3.1 (RadonNikodym theorem.) Let (Q,A,,u) be a U—ﬁnite measure space, and let
1/ be a measure on (Q,A) With 1/ << 11. Then there exists a measurable function X Z 0 such that
1/(A) : L, Xu for all A E A. The function X E Ell—:1 is unique in the sense that if Y is another such
function, then Y : X a.e. With respect to 11. X is called the Radon—Nikodym derivative of 1/ With
respect to ,u. Proof. See Billingsley (1986), page 376. Corollary 1 (Change of Variable Theorem.) Suppose that 1/, ,u are U—ﬁnite measures deﬁned
on a measure space (Q,A) With 1/ << 11, and suppose that Z is a measurable function such that f Zdz/ is well—deﬁned. Then for all A E A, Proof. If Z : 113; then d d
/1Bd1/:1/(AﬂB):/ —Vd,u:/ lB—Vdu.
A AnB am A dM (ii) If Z : 2,114,, then /Zd1/ : Za/lAidz/
A 12 CHAPTER 0. MEASURES, INTEGRATION, CONVERGENCE (iii) If Z Z 07 let Zn 2 0 be simple functions / Z. Then / Zdz/ : lim anI/ by the monotone convergence thm.
A d
lim/Znédu by part (ii)
: Z—du by the monotone convergence thm. (iv) If Z is measurable, Z : Z+ — Z‘ Where one of Z‘l’7 Z‘ is 1/—integrable7 then /Zdz/ : Z"dz/— Z_d1/
A A A
d1/ d1/
: Z"—d — Z —d b iii
A d” M A d” M y( )
: Zd—Vd
A dM Example 3.1 Let (SLA, P) be a probability space; often this Will be (1271,1871, P). Often in statis—
tics we suppose that P has a density f With respect to a U—ﬁnite measure ,u on (SLA) so that HAP/Ami for AEA. If ,u is Lebesgue measure on R”7 then f is the density function. If ,u is counting measure on a
countable set7 then f is the frequency function or mass function. Proposition 3.1 (Scheffé’s theorem.) Suppose that l/n(A) : fA fndn7 that l/(A) : fA fdu
Where fn are densities and l/n(9) : l/(Q) < 00 for all n, and that fn —> f a.e. M. Then sup WA) — M)! = lfn — fl a 0. AeA Proof. For A E A, WA) — um»  [4m — new Alfn—fldMS/Qlfn—flduv  /\ and this implies that
sup WA) — V(A) : /Q lfn — fldu AeA Let gn E f — Now g; —> 0 a.e. ,u, and g; g f Which is integrable. Thus by the dominated
convergence theorem ff]ng —> 0. But 0:/gndu=4(f—fn)du=é(gi—g;)dua 3. ABSOLUTE CONTINUITY, RADON—NIKODYM THEOREM, FUBINI’S THEOREM 13 so f5]ng : fggdu, and hence /lgnldu=/9$dM+/g;du=2/g$dM—>07 proving the claimed convergence. To prove that equality holds as claimed in the statement of the
proposition, note that for the event B E — fn Z 0] we have 31g)“ IMA) — V(A) Z IVn(B) — V(B) = [f_fn20](fn — f)dHl
= /[g;20]g$du = /9$du
= é/Ifn — fldu
But on the other hand
hMm—WMI=Ahw—Aﬂm
=Ijgf—nwm
= I AnB(f—ﬁadu—+ AOBgf—jtynd  /\ /g$dm sumacn—mAns/mwu=§/Ut—ﬂwu AeA SO Now supppose that (X,X,,u) and (Y,)},V) are two U—ﬁnite measure spaces. If A 6 X7 B E )1, a measurable rectangle is a set of the form A X B C X X Y.
Let X X 312 0({A X B : A E X,B E Deﬁne ameasure 7r on (XX Y,X>< )1) by mAxm=uwww>
for measurable rectangles A X B. Theorem 3.2 (Fubini  Tonelli theorem.) Suppose that f : X X Y —> R is X X y—measurable
and f 2 0. Then / f(x,y)dz/(y) is X — measurable ,
Y / f(9073/)d,u(96) is )1 — measurable ,
X and (u Aﬂjﬂwwﬂaw=é{éﬂawWWﬁwmﬂ=4{éﬂ%wWWﬁWW)
If f E L1(7r) (so [Xxy fd7r < oo)7 then (1) holds. ...
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