On Montel's Theorem release_dwn5vrkx4bfyrka6leiw4owvq4

by Yoshiro Kawakami

Published in Nagoya mathematical journal by Cambridge University Press (CUP).

1956   Volume 10, p125-127

Abstract

In this note we shall prove a theorem which is related to Montel's theorem [1] on bounded regular functions. Let <jats:italic>E</jats:italic> be a measurable set on the positive <jats:italic>y</jats:italic>-axis in the <jats:italic>z</jats:italic>( = <jats:italic>x</jats:italic> + <jats:italic>iy</jats:italic>)-plane, <jats:italic>E(a, b</jats:italic>) be its part contained in 0 <jats:italic>≦ a ≦ y ≦ b</jats:italic>, and ∣<jats:italic>E(a, b</jats:italic>)∣ be its measure. We define the lower density of <jats:italic>E</jats:italic> at <jats:italic>y</jats:italic> = 0 by <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0027763000000131_inline1" xlink:type="simple" /> L<jats:sc>EMMA</jats:sc>, <jats:italic>Let E be a set of positive lower density λ at y</jats:italic> = 0. <jats:italic>Then E contains a subset E<jats:sub>1</jats:sub> of the same lower density at y</jats:italic> = 0 <jats:italic>such that E<jats:sub>1</jats:sub> </jats:italic> ∪ {0} <jats:italic>is a closed set</jats:italic>.
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