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A Remark on the Cantor-Lebesgue Lemma
A REMARJC ON THE CANTOR-LEBESGUE LEMMA W.A. J. LUXEMBURGI) CaIvornia Institute of Technology
1. Introduction. One of the many applications of nonstandard methods given by Robinson in his book on Non-standard Analysis is a new and more straightforward proof of the famous Cantor-Lebesgue Lemma (see Hardy and Rogosinski [1950] p. 84 and Robinson [1966] Theorem 5.24, p. 132), which states that if lim (a, cos nx n+ca
+ b, sin nx) = 0
for all x in a set of positive measure, then limn-tma, = limn-rco b, = 0. Independently and at about the same time the present author also found a nonstandard proof of the Cantor-Lebesgue lemma. Although the author’s proof strongly resembles Robinson’s proof it differs from it in the sense that it is based on a convenient nonstandard formulation of a property of points of density one of sets measurable in the sense of Lebesgue. The purpose of the present paper is to present the essentials of a nonstandard formulation of Lebesgue’s theory of points of density of measurable sets and to show how it can be applied to obtain a generalization of the Cantor-Lebesgue Lemma.
2. Points of density and dispersion. For terminology and notation concerning nonstandard analysis not explained in the present paper we refer the reader to Robinson [1966] and Luxemburg [1972]. Let e be a Lebesgue measurable subset of the real line R. For the sake of convenience the characteristic function of e will also be denoted by the same letter e. For each measurable subset e c R we set E(x) = j: e(t)dt, where x, c E R and c is kept fixed. Points of density one of e and points of dispersion of e can be characterized as follows (see Saks [1948] Chap. IV, section 10, p. 128). 1)
This work was supported in part by NSF Grant GP 23392.
W.A. J. LUXEMBURG
42
2.1. A point x E R is a point of density one of a measurable subset e c R if and only if E is differentiable at x and its derivative E ( x ) = 1. A point x E R is a point of dispersion of e if and only if E'(x) = 0.
The reader should observe that a point x is a point of dispersion of a measurable set e if and only if it is a point of density one of the complement of e. In view of this fact we shall concentrate our attention to points of density one. The set of points of density one of a measurable subset e c R will be denoted by el. Concerning the set el we have the following result (see Saks [1948]). 2.2. For every measurable subset e c R the set el is measurable and &\el) + &1\e) = 0.
Consider now a nonstandard model of R . The field of nonstandard reals in the model will be denoted as usual by *R, the set of infinitesimals of *R by M , and the set of finite elements of * R by M,. The set of natural numbers (1, 2, ..., n, ...} of R will be denoted by N and in *R by *N. If a, b E *R, then a = b means that a - b is an infinitesimal. Our method in obtaining Cantor-Lebesgue type theorems is based on following property of points of density one in *R. 2.3. THEOREM. Let e be a measurable subset of R of positive measure and let xo E el be a point of density one of e. Thenfor each in$nitely large number y E *R\R and for each non-zerojinite standard real number a E Mo there exists a non-zero injinitesimal h (0 # h E M , ) such that yh = 1 a and xo rfs h E *el n *e. Proof. There is no loss in generality to assume that y and a of the theorem satisfy 0 < a and 0 < y. Let k be an infinitesimal such that 0 < k < a and let xo E el. Then from 0 < a/y E M 1and 2.1 it follows that
(
2.4. (i) ?- ( * E xo
k
+
e)
(
- *E xo +
Y
Hence, by adding the expressions 2.4 (i) and 2.4 (ii) and dividing by 2 we obtain
A REMARK ON THE CANTOR-LEBESGUE LEMMA
2.5.
2k
( * E (x,
43
+ +) - *E ( x o + -
-))
- *E ( x , - a - k Y
=ll.
It follows from 2.5 that there is an infinitesimalh # 0 such that xo f h Eel ne and a - k Ihy Ia. For otherwise the expression in *E in 2.5 would be at most 3, contradicting 2.5, and the proof is finished. An interesting class of real functions of a real variable, the so-called approximate continuous functions, was introduced by A. Denjoy (see Saks El9481 p. 131). We shall recall the definition. A measurable real functionj defined on an open interval I is called approximately continuous at a point xo E I whenever for every E > 0 the point xo is a point of density one of the set {x:If ( x ) - f(xo)l < E } . It was shown by A. Denjoy that approximately continuous functions on R are of Baire class 1 and have the Darboux property. Furthermore, Denjoy showed that iff is a real measurable function defined on a measurable set e, then f is almost everywhere approximately continuous on e. In the case that f is also bounded, then f is approximately continuous at a point x E e if and only if the function F(x) = J: f(t)dt is direrentiable at x and its derivative F ( x ) at x satisfies F ( x ) = f (x). From the last result we conclude that if e is the characteristic function of a measurable subset e of R , then e is approximately continuous at x i f and only if x is a point of density one of e. In view of Theorem 2.3, the reader will have no difficulty in proving the following theorem.
2.6. THEOREM. Let f be a real measurable function defined on R. Then for each infinitely large y E *R\Rand for each non-zerofinite number a(O # a EM,) there exists for almost all x E R an infinitesimal h = h,(x, y , a ) such that h # 0, hy = a and *f(xo h ) = f(xo). 3. A generalization of the Cantor-Lebesgue Lemma. Before we introduce a more general Cantor-Lebesgue Lemma we shall first present a proof of the Cantor-Lebesgue Lemma based on Theorem 2.3.
+
3.1. THEQREM (Cantor-Lebesgue Lemma). IfA,,(x) = a,, cos nx b,, sin nx = p,, cos(nx p,,), where p,, = ,/{a: b:), satisfies A,(x) = 0
+
+
W. A. J. LUXEMBURG
44
for all x in a set e ofpositive measure, then limn-rmpn = 0. Ifwe only assume that A, is uniformly bounded in n in a set of positive measure e, then the sequence {p,} is bounded. Proof. We shall first prove the limit theorem. From Egoroffs theorem it follows that there is no loss in generality to assume that limn+,mA,(x) = 0 uniformly in e. Hence, A,(z) = 0 for all z E *e and for all infinitely large natural numbers o E *N\N.Let xo E el n e be a point of density one of e and let w E *N\N. If cos(oxo + *pa) # 0, then *pa, = 0. If, on the other hand, cos(oxo + *p,) = 0, then by 2.3, there exists an infinitesimal h # 0 such that xo & h E *e and h o = 3n. Hence, 1 = 1 *po cos(o(xo h ) + *po) = 1 *pa, sin(oxo + *po) = 1 *po,since cos(oxo + *po) = 1 0 implies that sin(oxo + *pa) = 1 1. We conclude that *pa, = 1 0 for all o E *N\N which is equivalent to limn+- pn = 0, and finishes the proof of the fist part of the theorem. For the second part assume that IA,(x)l I M for all n E N and for all x E e. In order to show that {p,} is bounded we have only to show that*p, is finite for all o E *N\N. To this end, let xo E el n e and let o E *N\N. If cos(oxo + *po) = 0, then as above there exists an 0 # h E M l such that cos(o(xo + h ) + *,urn)= 1. From this we conclude that *po is finite for all o E: *N\N and the proof is finished.
+
From the various other applications one can make of Theorem 2.3 we have chosen to prove the following Cantor-Lebesgue Lemma type theorem. 3.2. THEOREM. Let f be a real periodic function of period one which is continuous everywhere and which has the property that its set of zeros has measure zero. Let {a,} (n E N ) be a sequence of real numbers such that limn-rma, = + co and assume that (8,) (n E N ) is an arbitrary sequence of real numbers. Then we have the following two results. (i) If {p,} (n E N ) is a sequence of real numbers such that
lim p,f(a,x n-rm
+ P,)
=0
in a measurable set e of positive measure, then limn-rmp, = 0.
(ii) If {p,} (n E N ) is a sequence of real numbers such that l~nf(anx+ PdI I M
for all n E N and for all x in a measurable set e of positive measure, then the sequence {p,} (n E N ) is bounded.
A REMARK ON THE CANTOR-LEBESGUE LEMMA
45
Proof. We shall only prove (i) since the proof of (ii) is similar and is, in fact, a little easier. As in the proof of the Cantor-Lebesgue Lemma there is no loss in generality to assume that limn+, pJ(a,,x 3/), = 0 uniformly in x E e. Hence, *p,*f(*ad */I,) = 0 for all o E *N\N and for all x E *e. Let xo E e n el be a point in e of density one and let o E *N\N. If *f(*a,xo + */I,) = 0, then since the set of zeros off has measure zero and e has positive measure it follows that there is a pointy E e such that If(y)i > 0. Let a E Mo be such that *a,xo */3, [*a,xo */?,I a = y. Sincelim,,, a,, = + 00 it follows that *a, is infinitely large, and so, by taking y = *a, in Theorem 2.3 we obtain that there exists an infinitesimalh # 0 such that *a,h = a and xo h E *e n *el. Hence, 0 = *p,*f(*a,(x, h) */Im) = *p,*f(y + k), where k E M , . Since If@)! > 0 and f is continuous at y we have that k ) = 0 implies that *pa = 0 for all *f(y + k ) # 0, and so *p,*f(y o E *N\N which finishes the proof.
+
+
+
-
+
+
+ +
+
+
Remark. If we take f ( x ) = cos 2nx (x E R) and a, obtain the Cantor-Lebesgue Lemma.
=
4 2 n (n E N), then we
From the above proof of Theorem 3.2 it follows that it is sufficient to assume that f is continuous almost everywhere. We shall now show that the conditions of the theorem are best possible. Let C be the Cantor set of the interval 0 Ix 5 1, and let f ( x ) = max (Ix yI:y E C ) 0 Ix I1 and extended periodically mod. 1. Then f is continuous and its set of zeros has measure zero. Since 3"x - [3"x] E C for all n E N and for all x E C we have, by taking p,, = n, for all n E N that limn+, p,f(3"x) = 0 for all x E C but limn+, p, = + 00, which shows that the hypothesis p(e) > 0 in (i) and (ii) of the theorem cannot be dropped. On the other hand, iff(x) = 0 for 0 Ix I3 andf(x) = (1 - 2x)(1 - x ) for t Ix I 1 and f ( x + 1) = f ( x ) for all other values of x, then f is continuous and its set of zeros in 0 Ix I 1 has measure 3. If p, = n (n E N ) , then, by observing that f(2"x) = 0 for all 0 5 x I3, we obtain that limn+, pJ(2,x) = 0 for all x in the set of positive measure 0 Ix I3 but the sequence {p,} is not even bounded, which shows that the condition that f may vanish only on a set of measure zero cannot be dropped in (i) and (ii) of the theorem. In conclusion we may point out that Theorem 2.3 can also be used to obtain a more straight forward proof of Kuttner's Theorem (see Hardy and Rogosinski [1950] p. 82) in the theory of trigonometric series.
-
46
W. A. J. LUXEMBURG
References Hardy, G. H. and Rogosinski, W. W., 1950, Fourier Series, Cambridge tracts in Mathematics and Mathematical Physics 38 (Cambridge, second ed.). Luxemburg, W. A. J., 1972, What is Nonstandard Analysis? Amer. Math. Monthly, to appear. Robinson, A., 1966, Non-Standard Analysis (North-Holland, Amsterdam). Saks, S., 1948, Theory of the Integral (New York, second rev. ed.). .Received 15 July 1971
1. Introduction. One of the many applications of nonstandard methods given by Robinson in his book on Non-standard Analysis is a new and more straightforward proof of the famous Cantor-Lebesgue Lemma (see Hardy and Rogosinski [1950] p. 84 and Robinson [1966] Theorem 5.24, p. 132), which states that if lim (a, cos nx n+ca
+ b, sin nx) = 0
for all x in a set of positive measure, then limn-tma, = limn-rco b, = 0. Independently and at about the same time the present author also found a nonstandard proof of the Cantor-Lebesgue lemma. Although the author’s proof strongly resembles Robinson’s proof it differs from it in the sense that it is based on a convenient nonstandard formulation of a property of points of density one of sets measurable in the sense of Lebesgue. The purpose of the present paper is to present the essentials of a nonstandard formulation of Lebesgue’s theory of points of density of measurable sets and to show how it can be applied to obtain a generalization of the Cantor-Lebesgue Lemma.
2. Points of density and dispersion. For terminology and notation concerning nonstandard analysis not explained in the present paper we refer the reader to Robinson [1966] and Luxemburg [1972]. Let e be a Lebesgue measurable subset of the real line R. For the sake of convenience the characteristic function of e will also be denoted by the same letter e. For each measurable subset e c R we set E(x) = j: e(t)dt, where x, c E R and c is kept fixed. Points of density one of e and points of dispersion of e can be characterized as follows (see Saks [1948] Chap. IV, section 10, p. 128). 1)
This work was supported in part by NSF Grant GP 23392.
W.A. J. LUXEMBURG
42
2.1. A point x E R is a point of density one of a measurable subset e c R if and only if E is differentiable at x and its derivative E ( x ) = 1. A point x E R is a point of dispersion of e if and only if E'(x) = 0.
The reader should observe that a point x is a point of dispersion of a measurable set e if and only if it is a point of density one of the complement of e. In view of this fact we shall concentrate our attention to points of density one. The set of points of density one of a measurable subset e c R will be denoted by el. Concerning the set el we have the following result (see Saks [1948]). 2.2. For every measurable subset e c R the set el is measurable and &\el) + &1\e) = 0.
Consider now a nonstandard model of R . The field of nonstandard reals in the model will be denoted as usual by *R, the set of infinitesimals of *R by M , and the set of finite elements of * R by M,. The set of natural numbers (1, 2, ..., n, ...} of R will be denoted by N and in *R by *N. If a, b E *R, then a = b means that a - b is an infinitesimal. Our method in obtaining Cantor-Lebesgue type theorems is based on following property of points of density one in *R. 2.3. THEOREM. Let e be a measurable subset of R of positive measure and let xo E el be a point of density one of e. Thenfor each in$nitely large number y E *R\R and for each non-zerojinite standard real number a E Mo there exists a non-zero injinitesimal h (0 # h E M , ) such that yh = 1 a and xo rfs h E *el n *e. Proof. There is no loss in generality to assume that y and a of the theorem satisfy 0 < a and 0 < y. Let k be an infinitesimal such that 0 < k < a and let xo E el. Then from 0 < a/y E M 1and 2.1 it follows that
(
2.4. (i) ?- ( * E xo
k
+
e)
(
- *E xo +
Y
Hence, by adding the expressions 2.4 (i) and 2.4 (ii) and dividing by 2 we obtain
A REMARK ON THE CANTOR-LEBESGUE LEMMA
2.5.
2k
( * E (x,
43
+ +) - *E ( x o + -
-))
- *E ( x , - a - k Y
=ll.
It follows from 2.5 that there is an infinitesimalh # 0 such that xo f h Eel ne and a - k Ihy Ia. For otherwise the expression in *E in 2.5 would be at most 3, contradicting 2.5, and the proof is finished. An interesting class of real functions of a real variable, the so-called approximate continuous functions, was introduced by A. Denjoy (see Saks El9481 p. 131). We shall recall the definition. A measurable real functionj defined on an open interval I is called approximately continuous at a point xo E I whenever for every E > 0 the point xo is a point of density one of the set {x:If ( x ) - f(xo)l < E } . It was shown by A. Denjoy that approximately continuous functions on R are of Baire class 1 and have the Darboux property. Furthermore, Denjoy showed that iff is a real measurable function defined on a measurable set e, then f is almost everywhere approximately continuous on e. In the case that f is also bounded, then f is approximately continuous at a point x E e if and only if the function F(x) = J: f(t)dt is direrentiable at x and its derivative F ( x ) at x satisfies F ( x ) = f (x). From the last result we conclude that if e is the characteristic function of a measurable subset e of R , then e is approximately continuous at x i f and only if x is a point of density one of e. In view of Theorem 2.3, the reader will have no difficulty in proving the following theorem.
2.6. THEOREM. Let f be a real measurable function defined on R. Then for each infinitely large y E *R\Rand for each non-zerofinite number a(O # a EM,) there exists for almost all x E R an infinitesimal h = h,(x, y , a ) such that h # 0, hy = a and *f(xo h ) = f(xo). 3. A generalization of the Cantor-Lebesgue Lemma. Before we introduce a more general Cantor-Lebesgue Lemma we shall first present a proof of the Cantor-Lebesgue Lemma based on Theorem 2.3.
+
3.1. THEQREM (Cantor-Lebesgue Lemma). IfA,,(x) = a,, cos nx b,, sin nx = p,, cos(nx p,,), where p,, = ,/{a: b:), satisfies A,(x) = 0
+
+
W. A. J. LUXEMBURG
44
for all x in a set e ofpositive measure, then limn-rmpn = 0. Ifwe only assume that A, is uniformly bounded in n in a set of positive measure e, then the sequence {p,} is bounded. Proof. We shall first prove the limit theorem. From Egoroffs theorem it follows that there is no loss in generality to assume that limn+,mA,(x) = 0 uniformly in e. Hence, A,(z) = 0 for all z E *e and for all infinitely large natural numbers o E *N\N.Let xo E el n e be a point of density one of e and let w E *N\N. If cos(oxo + *pa) # 0, then *pa, = 0. If, on the other hand, cos(oxo + *p,) = 0, then by 2.3, there exists an infinitesimal h # 0 such that xo & h E *e and h o = 3n. Hence, 1 = 1 *po cos(o(xo h ) + *po) = 1 *pa, sin(oxo + *po) = 1 *po,since cos(oxo + *po) = 1 0 implies that sin(oxo + *pa) = 1 1. We conclude that *pa, = 1 0 for all o E *N\N which is equivalent to limn+- pn = 0, and finishes the proof of the fist part of the theorem. For the second part assume that IA,(x)l I M for all n E N and for all x E e. In order to show that {p,} is bounded we have only to show that*p, is finite for all o E *N\N. To this end, let xo E el n e and let o E *N\N. If cos(oxo + *po) = 0, then as above there exists an 0 # h E M l such that cos(o(xo + h ) + *,urn)= 1. From this we conclude that *po is finite for all o E: *N\N and the proof is finished.
+
From the various other applications one can make of Theorem 2.3 we have chosen to prove the following Cantor-Lebesgue Lemma type theorem. 3.2. THEOREM. Let f be a real periodic function of period one which is continuous everywhere and which has the property that its set of zeros has measure zero. Let {a,} (n E N ) be a sequence of real numbers such that limn-rma, = + co and assume that (8,) (n E N ) is an arbitrary sequence of real numbers. Then we have the following two results. (i) If {p,} (n E N ) is a sequence of real numbers such that
lim p,f(a,x n-rm
+ P,)
=0
in a measurable set e of positive measure, then limn-rmp, = 0.
(ii) If {p,} (n E N ) is a sequence of real numbers such that l~nf(anx+ PdI I M
for all n E N and for all x in a measurable set e of positive measure, then the sequence {p,} (n E N ) is bounded.
A REMARK ON THE CANTOR-LEBESGUE LEMMA
45
Proof. We shall only prove (i) since the proof of (ii) is similar and is, in fact, a little easier. As in the proof of the Cantor-Lebesgue Lemma there is no loss in generality to assume that limn+, pJ(a,,x 3/), = 0 uniformly in x E e. Hence, *p,*f(*ad */I,) = 0 for all o E *N\N and for all x E *e. Let xo E e n el be a point in e of density one and let o E *N\N. If *f(*a,xo + */I,) = 0, then since the set of zeros off has measure zero and e has positive measure it follows that there is a pointy E e such that If(y)i > 0. Let a E Mo be such that *a,xo */3, [*a,xo */?,I a = y. Sincelim,,, a,, = + 00 it follows that *a, is infinitely large, and so, by taking y = *a, in Theorem 2.3 we obtain that there exists an infinitesimalh # 0 such that *a,h = a and xo h E *e n *el. Hence, 0 = *p,*f(*a,(x, h) */Im) = *p,*f(y + k), where k E M , . Since If@)! > 0 and f is continuous at y we have that k ) = 0 implies that *pa = 0 for all *f(y + k ) # 0, and so *p,*f(y o E *N\N which finishes the proof.
+
+
+
-
+
+
+ +
+
+
Remark. If we take f ( x ) = cos 2nx (x E R) and a, obtain the Cantor-Lebesgue Lemma.
=
4 2 n (n E N), then we
From the above proof of Theorem 3.2 it follows that it is sufficient to assume that f is continuous almost everywhere. We shall now show that the conditions of the theorem are best possible. Let C be the Cantor set of the interval 0 Ix 5 1, and let f ( x ) = max (Ix yI:y E C ) 0 Ix I1 and extended periodically mod. 1. Then f is continuous and its set of zeros has measure zero. Since 3"x - [3"x] E C for all n E N and for all x E C we have, by taking p,, = n, for all n E N that limn+, p,f(3"x) = 0 for all x E C but limn+, p, = + 00, which shows that the hypothesis p(e) > 0 in (i) and (ii) of the theorem cannot be dropped. On the other hand, iff(x) = 0 for 0 Ix I3 andf(x) = (1 - 2x)(1 - x ) for t Ix I 1 and f ( x + 1) = f ( x ) for all other values of x, then f is continuous and its set of zeros in 0 Ix I 1 has measure 3. If p, = n (n E N ) , then, by observing that f(2"x) = 0 for all 0 5 x I3, we obtain that limn+, pJ(2,x) = 0 for all x in the set of positive measure 0 Ix I3 but the sequence {p,} is not even bounded, which shows that the condition that f may vanish only on a set of measure zero cannot be dropped in (i) and (ii) of the theorem. In conclusion we may point out that Theorem 2.3 can also be used to obtain a more straight forward proof of Kuttner's Theorem (see Hardy and Rogosinski [1950] p. 82) in the theory of trigonometric series.
-
46
W. A. J. LUXEMBURG
References Hardy, G. H. and Rogosinski, W. W., 1950, Fourier Series, Cambridge tracts in Mathematics and Mathematical Physics 38 (Cambridge, second ed.). Luxemburg, W. A. J., 1972, What is Nonstandard Analysis? Amer. Math. Monthly, to appear. Robinson, A., 1966, Non-Standard Analysis (North-Holland, Amsterdam). Saks, S., 1948, Theory of the Integral (New York, second rev. ed.). .Received 15 July 1971