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II. A. S. Besicovitch and G. Walker, On the density of irregular linearly measurable sets of points. Proc. of London Math. Soc. (L. M. S.)32 (1931), pp. 142–153. III. J. Gillis, On linearly measurable plane sets of points of upper density 1/2. Fund. Math.22, pp. 57–70. IV. J. Gillis, Note on the projection of irregular linearly measurable plane sets of points. Fund. Math.26, pp. 229–233. V. J. Gillis, A Theorem on irregular linearly measurable sets of points. Journal of L. M. S. 10, pp. 234–240. VI. G. W. Morgan, The density directions of irregular linearly measurable plane sets. Proc. of L. M. S.38 (1935), pp. 481–494. We shall refer the cited papers by the Roman figures standing in front of them.
When talking of measurable sets we shall always mean sets of finite measure unless the opposite is stated.
Writing the product of sets we shall often omit the sign × for convenience of printing.
I, § 11, pp. 431–434.
Δ (A, B) denotes the distance between the setsA andB, so that Δ (a 0,a) is the distance between the pointsa 0 anda, Δ (ψ,a) is the distance from the curve ψ to the pointa, and so on. — u. bd = upper bound.
For a proof see R. Courant and D. Hilbert, Methoden der mathematischen Physik, Bd. I, Kap. II, § 2.
We denote byE 1-E 2 the set of points ofE 1 which do not belong toE 2;E 2 may or may not be entirely contained inE 1.
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I. A. S. Besicovitch, Paper under the same title, Math. Annalen98 (1927)
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Besicovitch, A.S. On the fundamental geometrical properties of linearly measurable plane sets of points (II). Math. Ann. 115, 296–329 (1938). https://doi.org/10.1007/BF01448943
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DOI: https://doi.org/10.1007/BF01448943