A Converse of Lebesgue�s Density Theorem.

A Converse of Lebesgue�s Density Theorem. - 350 [VoL...

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Unformatted text preview: 350 [VoL 6, 78. A Converse of Lebesgue’s Density Theorem. By Shunji KAMETANI. Taga Higher Technical School, Ibaragi. (Comm. by S. K A , M.I.A., Oct 12, 1940.) I. The main object of the present note is to see that a converse of Lebesgue’s density theorem holds. We shall consider, for brevity, sets of points in a Euclidean plane R s only. But the results which will be obtained can obviously be ex- tended to spaces R of any number of dimensions. The Lebesgue outer measure of a setE in R will be denoted by EI. Let z be a point of R and Q an arbitrary closedsquare contain- ing z with sides parallel to the coordinate-axes. We shall denote by ](z, E) and D(z,E) the superior and the in- ferior limit respectively of the ratio QEI/IQ] as the diameter of Q tends to 0 or ]Qi-0, and shall callthem the upper and the lower density of E at z respectively. If they are equal to each other at then the common value will be called the density of E at z. The points at which the density of E are equal to 0 are termed points of dispersion for E. Itiswell known by Lebesgue’s density theorem that, f set of points are measurable, almost every point of its complementary set is a poin of dispersion for he given set. D II. We shall prove the followingtheorem which evidently contains a converse of the above proposition. Theorem 1. Let E be a point-set whose lower density is 0 at almos every point of the complementary se of E. Then the se E is mrable. Proof. We can obviously assume, without loss of generality, that the set E is bounded. Let G be a bounded open set containing E and a given positive number. From the assumption of the present...
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This note was uploaded on 04/26/2010 for the course MATH 0101 taught by Professor Student during the Spring '10 term at École Normale Supérieure.

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A Converse of Lebesgue�s Density Theorem. - 350 [VoL...

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