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The Measure of Plane Sets

Published online by Cambridge University Press:  24 October 2008

P. A. P. Moran
P. A. P. Moran
Affiliation:
External Ballistics Laboratory, Free School Lane, Cambridge
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Extract

We consider bounded sets in a plane. If X is such a set, we denote by Pθ(X) the projection of X on the line y = x tan θ, where x and y belong to some fixed coordinate system. By f(θ, X) we denote the measure of Pθ(X), taking this, in general, as an outer Lebesgue measure. The least upper bound of f (θ, X) for all θ we denote by M. We write sm X for the outer two-dimensional Lebesgue measure of X. Then G. Szekeres(1) has proved that if X consists of a finite number of continua,

Béla v. Sz. Nagy(2) has obtained a stronger inequality, and it is the purpose of this paper to show that these results hold for more general classes of sets.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 39 , Issue 1 , March 1943 , pp. 51 - 53
Copyright
Copyright © Cambridge Philosophical Society 1943

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References

REFERENCES

(1)Szekeres, G.Ein Problem über mehrere ebene Bereiche. Acta Litt. Sci. Szeged, 9 (1940); 247–52.Google Scholar
(2)Nagy, , Béla v., Sz.Über ein geometrisches Externalproblem. Acta Litt. Sci. Szeged, 9(1940), 253–7.Google Scholar
(3)Mazurkiewicz, and Saks, . Sur les projections d'un ensemble fermé. Fundamenta Mathematicae (8), 109–13.Google Scholar