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A Converse of Lebesgueâ��s Density Theorem.

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

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350 [VoL 6, 78. A Converse of Lebesgue’s Density Theorem. By Shunji KAMETANI. Taga Higher Technical School, Ibaragi. (Comm. by S. KA, 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 set E in R will be denoted by EI. Let z be a point of R and Q an arbitrary closed square 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/I Q] as the diameter of Q tends to 0 or ]Qi-0, and shall call them 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. It is well 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 following theorem 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.

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

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