Article contents
On Montel’s Theorem
Published online by Cambridge University Press: 22 January 2016
Extract
In this note we shall prove a theorem which is related to Montel’s theorem [1] on bounded regular functions. Let E be a measurable set on the positive y-axis in the z( = x + iy)-plane, E(a, b) be its part contained in 0 ≦ a ≦ y ≦ b, and ∣E(a, b)∣ be its measure. We define the lower density of E at y = 0 by
LEMMA, Let E be a set of positive lower density λ at y = 0. Then E contains a subset E1 of the same lower density at y = 0 such that E1 ∪ {0} is a closed set.
- Type
- Research Article
- Information
- Copyright
- Copyright © Editorial Board of Nagoya Mathematical Journal 1956
References
- 3
- Cited by