On Montel’s Theorem

Source abstract

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 L EMMA , Let E be a set of positive lower density λ at y = 0. Then E contains a subset E 1 of the same lower density at y = 0 such that E 1 ∪ {0} is a closed set .