Lebesgue\u27s density theorem states that at almost every point of a measurable set S in En, the metric density of S exists and is 1 and at almost every point of the complement of S, the density of S exists and is 0. This theorem was first proven for E1 by Lebesgue using his theory of integration. It was later proven by Denjoy [1], Lusin [2], and Sierpinski [3] for E1 without the use of integration. The theorem was first proven for En by de la Vallee Poussin

Martin, N. F. G.

English

University of Northern Iowa

Proceedings of the Iowa Academy of ScienceVolume 65 | Annual Issue Article 491958Exceptional Values of Metric DensityN. F. G. MartinIowa State CollegeCopyright © Copyright 1958 by the Iowa Academy of Science, Inc.Follow this and additional works at: https://scholarworks.uni.edu/piasThis Research is brought to you for free and open access by UNI ScholarWorks. It has been accepted for inclusion in Proceedings of the Iowa Academyof Science by an authorized editor of UNI ScholarWorks. For more information, please contact scholarworks@uni.edu.Recommended CitationMartin, N. F. G. (1958) "Exceptional Values of Metric Density," Proceedings of the Iowa Academy of Science: Vol. 65: No. 1 , Article 49.Available at: https://scholarworks.uni.edu/pias/vol65/iss1/49Exceptional Values of Metric Density By N. F. G. MARTIN Lebesgue's density theorem states that at almost every point of a measurable set S in En, the metric density of S exists and is 1 and at almost every point of the complement of S, the density of S exists and is 0. This theorem was first proven for E 1 by Lebesgue using his theory of integration. It was later proven by Denjoy [1], Lusin [2], and Sierpinski [ 3] for E1 without the use of integration. The the-orem was first proven for En by de la Vallee Poussin. In the light of the density theorem one might say that the "usual" values of metric density are 0 and 1. It is easy to give examples where the density does not exist. The purpose of this note is to show that the density in E1 may have any value between 0 and 1. Let {In} be a sequence of intervals in E 1. The sequence is said to converge to the real number x, if ( i) x is a member of In for each n and (ii) {m(In)}, where m denotes Lebesgue measure, is a null sequence. Let <P be a real valued function defined on a subclass I of the class of all intervals in E 1 . If {Ik} is a sequence of intervals from I, then lim sup <P (h) and lim inf cf> (lk) will denote the right most and left most limit points of the sequence { <P (h) }. Denote by I (x) the class consisting of all sequences of intervals from I which converge to x. Then lim sup <P (I) is defined to be sup {lim sup cf> (h): 1->x (h} (I (x) }, and lim inf <f> (I) is defined to be inf (lim inf <P (Ik): 1->x The relative measure of a measurable set S in an interval I, de-noted by p ( S: I), will mean the ratio of the measure of S ,......_ I to the measure of I. Some obvious properties of p(S:I) are the following: (1) For a given measurable set S, p(S:I) is defined for all bounded intervals in Ei. (2) 0 < p(S:I) < 1. (3) Ifm(S,......_I) =0,µ(S:I) =0. (4) If S ~I, µ(S:I) = 1. (5) If -S denotes the complement of S, then p(-S:l)=l-p(S:I). 335 1Martin: Exceptional Values of Metric DensityPublished by UNI ScholarWorks, 1958336 IOWA ACADEMY OF SCIENCE Statement (5) follows from the fact that m(S,.....,I) + m(-S,.....,I) = m(I). [Vol. 65 Now let S be any measurable set in E1 and let x be any real number. Then the upper metric density of Sat x, denoted by D~(S), is defined to be lim sup p ( S: I), and the lower metric density, I->x Dx (S) is defined to be lim inf p(S:I). I->x If Dx ( S) = :Q.x ( S), then the common value is called the metric density of Sat x and is denoted by !?x(S). By the definition of Dx(S) and !?x (S) and (2) it follows that (6) 0 < _!?x (S) < Dx (S) < 1. If Dx (S) exists, then (7) Dx (S) = lim p(S:Ik) where Ik - > x. From (7) and statement (5) it follows that if Dx (S) exists then (8) Dx (-S) = 1 - Dx (S). Example 1. Let S = { x: 0 < x < 1}. Then if O<x < 1, Dx ( S) =1, and if x > 1 or x < 0, then Dx(S) = 0. For the points 0 and 1, Dx(S) fails to exist. It is obvious that Dx(S) = 1 for points of S different from zero, and Dx(S) = 0 for points of -S different from 1. Let Ik = [- ~, O]. Then Ik - > 0 and p(S:h) = 0. Hence ~o (S) = 0. If Ik = [o .!. ], h - > 0 and p(S:Ik) = 1. Therefore Do (S) = 1, and ' k it follows that D0 (S) does not exist. In example 1, where Dx(S) = O, there is some interval, I, about x such that m(S,.....,I) = 0. An interesting example due essentially to Goffman [ 4], is the following in which D0 (S) = 0 but for each open interval, I, containing 0, m(S,--1) > 0. Example 2. For each positive integer n, let I 1 1 1 } An= l x: n< x < n + zn . 00 Then if S = U An, D 0 (S) = 0. For, if I is any interval about n=l O, and if Jn= [ O, _!__l ] and Kn= [ 0, .!_ + 21 n ], there is an n- n 2Proceedings of the Iowa Academy of Science, Vol. 65 [1958], No. 1, Art. 49https://scholarworks.uni.edu/pias/vol65/iss1/491958] METRIC DENSITY 337 n such that Therefore D 0 (S) < lim m(S,....._Kn) . 00 1 1 1 ·- =hm :S -k/-+-n=O. Kn-> 0 m(Kn) n-> ook=n 2 n 2 The following example gives a set whose density exists at 0 and has any given value between 0 and 1. Example 3. Let 0 < ,\ < 1 be given. For each positive integer n, let Au= L .. ~ Ln and Bn = M 11 ~ .M" where Ln = ix: n+l < x < n+l + n(n~l) ~ I 1 ,\ 1 l Ln = l x: - n+1 n(n+l) < x < - n+1 f Mn= { x: n!l + n(n~l) < x < ~ } - { 1 1 ,\ ) Mn= x: -n < x < - n+l - n(n+l) 1· 00 00 Then let A= U An and B = U Bn. Then D0 (A) = ,\. n=l n=l Proof. Let I be any interval containing 0 whose length is less than 1/2. Denote the left and right end points of I by h and k respectively. Since m(I) < 1, there exists an integer N such that 1 1 (9) N+l < m(I) < N-· Also there exist integers p and q such that (10) 1 1 (11) - q-l·<h< q 1 1 Let IR= ( -, k) and 11, = (h, - -· ). Then p q m(IR) = k - -~-< p(;-l) and m(h) < -q(-q-1- 1-) 3Martin: Exceptional Values of Metric DensityPublished by UNI ScholarWorks, 1958338 IOWA ACADEMY OF SCIENCE From inequalities (9) and (10) 1 1 ~ < k < m(I) < --p N [Vol. 65 andp > N. Thusp(p~l) <N(~-i) .Itfollowsfrominequalities ( 9) and ( 11) in a similar manner that ( 1 ) q q-1 Therefore, if E = In~h, m(E) < N(~-1) Again using inequality (9) it follows that (lZ) m(E) < 2(N+l). m(I) - N(N-1) 1 < N(N-1) Let H = (A'""""'I) ~ (B'""""'I) - E. Then H consists of disjoint open intervals L0 , Mn; n = p, p+l, ... and Lm, Mm; m = q, q+ 1, . . . The interval I may be written as (13) I= H~E~D, where D is a countable set consisting of the point 0 and the end points of the disjoint intervals in H. Since H, E, and D are disjoint, (14) m(I) = m(H) + m(E). Now, 00 00 (15) ~ 1 m(H) = n=p n(n+l) ~ 1 + n=q n(n+l) _!_, + .!. p q and oo oo (16) m(A'""""'H) = ~ m(L0 ) + ~ m(Ln) n=p n=q 00 00 ~ n=p = A( .!_ p Therefore p(A:H) =A. A n(n+l) + .!_ ). q From equation ( 14) it follows that m(A'""""'H) < . _ (17) m(I) _ p(A.H)-A. It is also true that (18) m(A'""""'H) = Am(H)/m(I), m(I) ~ A + m=q m(m+l) 4Proceedings of the Iowa Academy of Science, Vol. 65 [1958], No. 1, Art. 49https://scholarworks.uni.edu/pias/vol65/iss1/491958] METRIC DENSITY 339 but division of equation (14) by m(I) and rearrangement of terms gives m(H) _ 1 _ m(E) . m(I) - m(I) It then follows from inequality (12) that (l 9) m(H) > 1 _ 2(N+l). m(I) - N(N-1) Combining inequalities (17), (18), and (19) gives that (20) , 2A(N+l) < m(A~H) < , " - N(N-1) m(I) - "· From equation ( 13) (21) m(A~I) = m(A~H) + m(A~E). Therefore m(A~I) m(I) and = m(A~H) + m(I) + 2(N+l) < A. N(N-1)' m(A~E) m(I) m(A~I) > m(I) m(A~H) m(I) Thus 2A(N+l). N(N-1) (22) A. - :.;-(~ll; < p(A:I) < A.+ 2(N+l). N(N-1) For any sequence Ir - > 0, the sequence N r of integers associated with Ir must approach co. Therefore by (22), D 0 (A) = ,.\. Literature Cited !. Denjoy, A., Journal de Math. (7) vol. 1 (1915) p. 132. 2. Lusin, N., Rend. d. Gire. Math de Palermo, vol. XLIV (1917). 3. Sierpinski, W., Fund. Math. vol. IV (1923). 4. Goffman, C., Real Functions, Rinehart and Co. (1953) p. 173. DEPARTMENT OF MATHEMATICS lowA STATE COLLEGE AMES, lowA 5Martin: Exceptional Values of Metric DensityPublished by UNI ScholarWorks, 1958

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Exceptional Values of Metric Density

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